  
  [1X4 [33X[0;0YSpherical and Projective Designs[133X[101X
  
  [33X[0;0YA  spherical  or  projective  design  is  a  finite  subset  of  a sphere or
  projective space (see [DGS77] and [Hog82] for more details). Certain designs
  have  special properties and interesting symmetries. The [5XALCO[105X package allows
  users  to  study  both spherical and projective designs by modelling both as
  finite sets of primitive idempotents of a simple Euclidean Jordan algebra.[133X
  
  [33X[0;0YSpecifically,  the primitive idempotents of simple Euclidean Jordan algebras
  of  rank  [23X2[123X  have  the  geometry  of  a  sphere. The correspondence involves
  converting  Euclidean inner product [23X\cos(\alpha)[123X between two unit vectors on
  a  sphere  into the corresponding Jordan inner product [23X\mathrm{Tr}(x\circ y)[123X
  given  by  [23X(1 + \cos(\alpha))/2[123X (described in [Nas23, p. 72]). Likewise, the
  primitive idempotents of a simple Euclidean Jordan algebras of degrees [23X1[123X, [23X2[123X,
  [23X4[123X,  or  [23X8[123X  have  the  geometry  of  a real, complex, quaternion, or octonion
  projective space.[133X
  
  [33X[0;0YThe  tools  below  allow one to construct a [5XGAP[105X object to represent a design
  and  collect  various  computed  attributes.  Constructing  a design and its
  parameters  using  these  tools  does  not guarantee the existence of such a
  design,  although known examples and possible instances may be studied using
  these tools.[133X
  
  
  [1X4.1 [33X[0;0YJacobi Polynomials[133X[101X
  
  [33X[0;0YOne  advantage of studying spherical and projective designs together as sets
  of  Jordan  primitive idempotents is both the spherical and projective cases
  can  be  studied together using Jacobi polynomials, with suitable parameters
  chosen for the appropriate simple Euclidean Jordan algebra.[133X
  
  [1X4.1-1 JacobiPolynomial[101X
  
  [33X[1;0Y[29X[2XJacobiPolynomial[102X( [3Xk[103X, [3Xa[103X, [3Xb[103X ) [32X function[133X
  
  [33X[0;0YThis  function  returns the Jacobi polynomial [23XP_k^{(a,b)}(x)[123X of degree [3Xk[103X and
  type  [3X(a,b)[103X  as  defined  in  [AS72,  chap.  22].  The  argument [3Xk[103X must be a
  non-negative  integer. The arguments [3Xa[103X and [3Xb[103X must be either rational numbers
  greater than [23X-1[123X or must satisfy [10XIsPolynomial[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa := Indeterminate(Rationals, "a");; [127X[104X
    [4X[25Xgap>[125X [27Xb := Indeterminate(Rationals, "b");; [127X[104X
    [4X[25Xgap>[125X [27Xx := Indeterminate(Rationals, "x");;[127X[104X
    [4X[25Xgap>[125X [27XJacobiPolynomial(0,a,b);[127X[104X
    [4X[28X[ 1 ][128X[104X
    [4X[25Xgap>[125X [27XJacobiPolynomial(1,a,b);[127X[104X
    [4X[28X[ 1/2*a-1/2*b, 1/2*a+1/2*b+1 ][128X[104X
    [4X[25Xgap>[125X [27XValuePol(last,x);[127X[104X
    [4X[28X1/2*a*x+1/2*b*x+1/2*a-1/2*b+x[128X[104X
  [4X[32X[104X
  
  
  [1X4.1-2 [33X[0;0YRenormalized Jacobi Polynomials[133X[101X
  
  [33X[1;0Y[29X[2XQ_k_epsilon[102X( [3Xk[103X, [3Xepsilon[103X, [3Xrank[103X, [3Xdegree[103X, [3Xx[103X ) [32X function[133X
  [33X[1;0Y[29X[2XR_k_epsilon[102X( [3Xk[103X, [3Xepsilon[103X, [3Xrank[103X, [3Xdegree[103X, [3Xx[103X ) [32X function[133X
  
  [33X[0;0YThese  functions  return  polynomials  of  degree  [3Xk[103X  in the indeterminate [3Xx[103X
  corresponding  the the renormalized Jacobi polynomials given in [Hog82]. The
  value of [3Xepsilon[103X must be 0 or 1. The arguments [3Xrank[103X and [3Xdegree[103X correspond to
  the rank and degree of the relevant simple Euclidean Jordan algebra.[133X
  
  
  [1X4.2 [33X[0;0YJordan Designs[133X[101X
  
  [33X[0;0YThe  [5XALCO[105X  package  defines  new  categories  within  [10XIsObject[110X  in  order to
  construct and study Jordan designs.[133X
  
  
  [1X4.2-1 [33X[0;0YJordan Design Categories[133X[101X
  
  [33X[1;0Y[29X[2XIsJordanDesign[102X [32X filter[133X
  [33X[1;0Y[29X[2XIsSphericalJordanDesign[102X [32X filter[133X
  [33X[1;0Y[29X[2XIsProjectiveJordanDesign[102X [32X filter[133X
  
  [33X[0;0YThese filters determine whether an object is a Jordan design and whether the
  design  is  constructed  in  a  spherical  or  projective manifold of Jordan
  primitive idempotents.[133X
  
  [1X4.2-2 JordanDesignByParameters[101X
  
  [33X[1;0Y[29X[2XJordanDesignByParameters[102X( [3Xrank[103X, [3Xdegree[103X ) [32X function[133X
  
  [33X[0;0YThis function constructs a Jordan design in the manifold of Jordan primitive
  idempotents of rank [3Xrank[103X and degree [3Xdegree[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByParameters(3,8);[127X[104X
    [4X[28X<design with rank 3 and degree 8>[128X[104X
    [4X[25Xgap>[125X [27XIsJordanDesign(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsSphericalJordanDesign(D);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsProjectiveJordanDesign(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByParameters(4,8);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByParameters(3,9);[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  
  [1X4.2-3 [33X[0;0YJordan Rank and Degree[133X[101X
  
  [33X[1;0Y[29X[2XJordanDesignRank[102X( [3XD[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XJordanDesignDegree[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YThe rank and degree of an object satisfying filter [10XIsJordanDesign[110X are stored
  as attributes.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByParameters(3,8);[127X[104X
    [4X[28X<design with rank 3 and degree 8>[128X[104X
    [4X[25Xgap>[125X [27X[JordanDesignRank(D), JordanDesignDegree(D)];[127X[104X
    [4X[28X[ 3, 8 ][128X[104X
  [4X[32X[104X
  
  [1X4.2-4 JordanDesignQPolynomials[101X
  
  [33X[1;0Y[29X[2XJordanDesignQPolynomials[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YMany  properties  of  a  Jordan  design  are  computed  using  the family of
  renormalized   Jacobi  polynomials  that  correspond  to  the  spherical  or
  projective   space   in   question.   This   attribute   stores  a  function
  [10XDesignQPolynomial([3XD[103X[10X)([3Xk[103X[10X)[110X that returns the [3Xk[103X-th polynomial in the family, as a
  list of coefficients, where [3Xk[103X is a non-negative integer.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByParameters(3,8);[127X[104X
    [4X[28X<design with rank 3 and degree 8>[128X[104X
    [4X[25Xgap>[125X [27Xr := JordanDesignRank(D);; d := JordanDesignDegree(D);;[127X[104X
    [4X[25Xgap>[125X [27Xx := Indeterminate(Rationals, "x");;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignQPolynomials(D);[127X[104X
    [4X[28Xfunction( k ) ... end[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignQPolynomials(D)(2);[127X[104X
    [4X[28X[ 90, -585, 819 ][128X[104X
    [4X[25Xgap>[125X [27XCoefficientsOfUnivariatePolynomial(Q_k_epsilon(2,0,r,d,x));[127X[104X
    [4X[28X[ 90, -585, 819 ][128X[104X
  [4X[32X[104X
  
  [1X4.2-5 JordanDesignConnectionCoefficients[101X
  
  [33X[1;0Y[29X[2XJordanDesignConnectionCoefficients[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YThe   connection   coefficients   of  a  design  [3XD[103X  determine  which  linear
  combinations   of   [10XJordanDesignQPolynomials([3XD[103X[10X)[110X  yield  each  power  of  the
  indeterminate   [Hog92,   p.   261].   This   attribute  stores  a  function
  [10XJordanDesignConnectionCoefficients([3XD[103X[10X)([3Xs[103X[10X)[110X   that   computes   the  connection
  coefficients of each power up to positive integer [3Xs[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByParameters(3,8);[127X[104X
    [4X[28X<design with rank 3 and degree 8>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignConnectionCoefficients(D);[127X[104X
    [4X[28Xfunction( s ) ... end[128X[104X
    [4X[25Xgap>[125X [27Xf := JordanDesignConnectionCoefficients(D)(3);; Display(f);[127X[104X
    [4X[28X[ [        1,        0,        0,        0 ],[128X[104X
    [4X[28X  [      1/3,     1/39,        0,        0 ],[128X[104X
    [4X[28X  [     5/39,    5/273,    1/819,        0 ],[128X[104X
    [4X[28X  [     5/91,     1/91,    1/728,  1/12376 ] ][128X[104X
    [4X[25Xgap>[125X [27Xfor j in [1..4] do Display(Sum(List([1..4], i -> [127X[104X
    [4X[25X>[125X [27Xf[j][i]*JordanDesignQPolynomials(D)(i-1)))); od;[127X[104X
    [4X[28X[ 1, 0, 0, 0 ][128X[104X
    [4X[28X[ 0, 1, 0, 0 ][128X[104X
    [4X[28X[ 0, 0, 1, 0 ][128X[104X
    [4X[28X[ 0, 0, 0, 1 ][128X[104X
  [4X[32X[104X
  
  
  [1X4.3 [33X[0;0YDesigns with an Angle Set[133X[101X
  
  [33X[0;0YThe  angle set of a design is the set of all inner products between distinct
  elements  in  the design. In the case of a Jordan design, each inner product
  is computed as [23X\mathrm{Tr}(x\circ y)[123X for [23Xx[123X and [23Xy[123X primitive idempotents. This
  means  that  the  angle  set  of  a  design  is a set of real numbers in the
  interval  [23X[0, 1)[123X. We can compute many additional properties of a design once
  the angle set is known.[133X
  
  [1X4.3-1 IsJordanDesignWithAngleSet[101X
  
  [33X[1;0Y[29X[2XIsJordanDesignWithAngleSet[102X [32X filter[133X
  
  [33X[0;0YThis  filter  identifies  whether an object that satisfies [10XIsJordanDesign[110X is
  equipped with an angle set.[133X
  
  
  [1X4.3-2 [33X[0;0YDesign Angle Sets[133X[101X
  
  [33X[1;0Y[29X[2XJordanDesignAddAngleSet[102X( [3XD[103X, [3XA[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XJordanDesignAngleSet[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YFor a design [3XD[103X without an angle set, records the angle set [3XA[103X as an attribute
  [10XJordanDesignAngleSet[110X. The angle set [3XA[103X must be a list of real-valued elements
  in  [10XIsCyc[110X  in  the  interval  [23X[0,  1)[123X.  Note that when [3XA[103X contains irrational
  elements for which < does not provide an ordering, inclusion in the interval
  given  above is not checked. Also note that the angle set cannot be modified
  once set as an attribute of the design.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByParameters(4,4);[127X[104X
    [4X[28X<design with rank 4 and degree 4>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddAngleSet(D, [2]);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XD;[127X[104X
    [4X[28X<design with rank 4 and degree 4>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddAngleSet(D, [1/3,1/9]);[127X[104X
    [4X[28X<design with rank 4, degree 4, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignAngleSet(D);[127X[104X
    [4X[28X[ 1/9, 1/3 ][128X[104X
  [4X[32X[104X
  
  [1X4.3-3 JordanDesignByAngleSet[101X
  
  [33X[1;0Y[29X[2XJordanDesignByAngleSet[102X( [3Xrank[103X, [3Xdegree[103X, [3XA[103X ) [32X function[133X
  
  [33X[0;0YThis  function  constructs a new design with Jordan rank and degree given by
  [3Xrank[103X  and [3Xdegree[103X, with angle set [3XA[103X. The same constrains on angle set [3XA[103X given
  in [2XJordanDesignAddAngleSet[102X ([14X4.3-2[114X) apply.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4, 4, [1/3, 1/9]);[127X[104X
    [4X[28X<design with rank 4, degree 4, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignAngleSet(D);[127X[104X
    [4X[28X[ 1/9, 1/3 ][128X[104X
  [4X[32X[104X
  
  [1X4.3-4 JordanDesignNormalizedAnnihilatorPolynomial[101X
  
  [33X[1;0Y[29X[2XJordanDesignNormalizedAnnihilatorPolynomial[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YThe  normalized  annihilator polynomial is defined for an angle set [23XA[123X as the
  polynomial [23Xp(x)[123X of degree equal to the cardinality of [23XA[123X with the elements of
  [23XA[123X  for  roots  and  normalization  such  that [23Xp(1) = 1[123X [BBIT21, p. 185]. The
  coefficients  of this polynomial are stored as an attribute of a design with
  an angle set.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4, 4, [1/3, 1/9]);[127X[104X
    [4X[28X<design with rank 4, degree 4, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27Xp := JordanDesignNormalizedAnnihilatorPolynomial(D);[127X[104X
    [4X[28X[ 1/16, -3/4, 27/16 ][128X[104X
    [4X[25Xgap>[125X [27XValuePol(p, 1/9);[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XValuePol(p, 1/3);[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XValuePol(p, 1);[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [1X4.3-5 JordanDesignNormalizedIndicatorCoefficients[101X
  
  [33X[1;0Y[29X[2XJordanDesignNormalizedIndicatorCoefficients[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YThe        normalized       indicator       coefficients       are       the
  [10XJordanDesignQPolynomials([3XD[103X[10X)[110X-expansion             coefficients            of
  [10XJordanDesignNormalizedAnnihilatorPolynomial([3XD[103X[10X)[110X,  discussed for the spherical
  case in [BBIT21, p. 185]. These coefficients are stored as an attribute of a
  design with an angle set.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4, 4, [1/3, 1/9]);[127X[104X
    [4X[28X<design with rank 4, degree 4, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27Xf := JordanDesignNormalizedIndicatorCoefficients(D);[127X[104X
    [4X[28X[ 1/64, 7/960, 9/3520 ][128X[104X
    [4X[25Xgap>[125X [27XSum(List([1..3], i -> f[i]*JordanDesignQPolynomials(D)(i-1)));[127X[104X
    [4X[28X[ 1/16, -3/4, 27/16 ][128X[104X
    [4X[25Xgap>[125X [27XJordanDesignNormalizedAnnihilatorPolynomial(D);[127X[104X
    [4X[28X[ 1/16, -3/4, 27/16 ][128X[104X
  [4X[32X[104X
  
  [1X4.3-6 IsJordanDesignWithPositiveIndicatorCoefficients[101X
  
  [33X[1;0Y[29X[2XIsJordanDesignWithPositiveIndicatorCoefficients[102X [32X filter[133X
  
  [33X[0;0YThis  filter  determines  whether the normalized indicator coefficients of a
  design  are  positive,  which  has  significance  for certain theorems about
  designs.[133X
  
  [1X4.3-7 JordanDesignSpecialBound[101X
  
  [33X[1;0Y[29X[2XJordanDesignSpecialBound[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YThe       special       bound       of       a       design       satisfying
  [10XIsJordanDesignWithPositiveIndicatorCoefficients[110X  is  the  upper limit on the
  possible  cardinality  for  the  given  angle set [Hog92, pp. 257-258]. This
  attribute stores the special bound when it exists for a design.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4, 4, [1/3,1/9]);[127X[104X
    [4X[28X<design with rank 4, degree 4, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XIsJordanDesignWithPositiveIndicatorCoefficients(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignSpecialBound(D);[127X[104X
    [4X[28X64[128X[104X
  [4X[32X[104X
  
  
  [1X4.4 [33X[0;0YDesigns with Angle Set and Cardinality[133X[101X
  
  [33X[0;0YMany  more properties of a design with an angle set can be computed once the
  cardinality  of  the  design  is  also  known.  In what follows let [23Xv[123X be the
  cardinality  of  a design and let [23Xs[123X be the cardinality of the angle set [23XA[123X of
  that design.[133X
  
  
  [1X4.4-1 [33X[0;0YDesign Cardinality[133X[101X
  
  [33X[1;0Y[29X[2XIsJordanDesignWithCardinality[102X [32X filter[133X
  [33X[1;0Y[29X[2XJordanDesignAddCardinality[102X( [3XD[103X, [3Xv[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XJordanDesignCardinality[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YAs  a  finite  set,  each design has a cardinality. When this cardinality is
  known  for  an object [3XD[103X that satisfies [10XIsJordanDesign([3XD[103X[10X)[110X, the cardinality is
  stored  as  the  attribute  [10XJordanDesignCardinality([3XD[103X[10X)[110X.  In order to set the
  cardinality     of     a     design,    we    can    use    the    operation
  [10XJordanDesignAddCardinality([3XD[103X[10X, v)[110X. When [10XJordanDesignAddCardinality[110X is called,
  the  [5XALCO[105X  package  immediately  attempts  to  compute  [2XJordanDesignStrength[102X
  ([14X4.4-5[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4,4, [1/3,1/9]);[127X[104X
    [4X[28X<design with rank 4, degree 4, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XHasJordanDesignCardinality(D);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(D, 64);[127X[104X
    [4X[28X<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignCardinality(D);[127X[104X
    [4X[28X64[128X[104X
  [4X[32X[104X
  
  
  [1X4.4-2 [33X[0;0YDesigns at the Special Bound[133X[101X
  
  [33X[1;0Y[29X[2XIsSpecialBoundJordanDesign[102X [32X filter[133X
  
  [33X[0;0YAs described in [2XJordanDesignSpecialBound[102X ([14X4.3-7[114X), we can compute the special
  bound of a design using the angle set. Once the cardinality is also known we
  can  assess  whether  the  design  reaches  the  special  bound. This filter
  identifies  when  a  design with an angle set and cardinality also meets the
  special bound.[133X
  
  [1X4.4-3 JordanDesignAnnihilatorPolynomial[101X
  
  [33X[1;0Y[29X[2XJordanDesignAnnihilatorPolynomial[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YThe  annihilator  polynomial  for  design  [3XD[103X  is  defined by multiplying the
  [10XJordanDesignNormalizedAnnihilatorPolynomial([3XD[103X[10X)[110X                            by
  [10XJordanDesignCardinality([3XD[103X[10X)[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4, 4, [1/3, 1/9]);; [127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(D, 64);; D;[127X[104X
    [4X[28X<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignAnnihilatorPolynomial(D);[127X[104X
    [4X[28X[ 4, -48, 108 ][128X[104X
    [4X[25Xgap>[125X [27XValuePol(last, 1);[127X[104X
    [4X[28X64[128X[104X
  [4X[32X[104X
  
  [1X4.4-4 JordanDesignIndicatorCoefficients[101X
  
  [33X[1;0Y[29X[2XJordanDesignIndicatorCoefficients[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YThe  indicator  coefficients  for  design  [3XD[103X  are  defined  by  multiplying [10X
  JordanDesignNormalizedIndicatorCoefficients([3XD[103X[10X)[110X                            by
  [10XJordanDesignCardinality([3XD[103X[10X)[110X.  These  indicator  coefficients are often useful
  for directly determining the strength of a design at the special bound.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4, 4, [1/3, 1/9]);; JordanDesignAddCardinality(D, 64);; D;[127X[104X
    [4X[28X<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignIndicatorCoefficients(D);[127X[104X
    [4X[28X[ 1, 7/15, 9/55 ][128X[104X
  [4X[32X[104X
  
  
  [1X4.4-5 [33X[0;0YDesign Strength[133X[101X
  
  [33X[1;0Y[29X[2XIsJordanDesignWithStrength[102X [32X filter[133X
  [33X[1;0Y[29X[2XJordanDesignStrength[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YThe  [23Xt[123X-design  is a design with the following special property: the integral
  of  any  degree  [23Xt[123X polynomial over the sphere or projective space containing
  the design is equal to the average value of that polynomial evaluated at the
  points  of  the [23Xt[123X-design (see [DGS77] and [Hog82] for detailed definitions).
  The parameter [23Xt[123X is called the [13Xstrength[113X of the design.[133X
  
  [33X[0;0YFor           a           design          [3XD[103X          that          satisfies
  [10XIsJordanDesignWithPositiveIndicatorCoefficients[110X,
  [10XIsJordanDesignWithCardinality[110X,   and   [10XIsSpecialBoundJordanDesign[110X,   we  can
  compute  the  strength  [23Xt[123X  of the design using a theorem given in [Hog92, p.
  258]    that    examines    the    indicator    coefficients.   The   filter
  [10XIsJordanDesignWithStrength[110X indicates when the attribute [10XJordanDesignStrength[110X
  has been successfully computed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4,4, [1/3,1/9]);[127X[104X
    [4X[28X<design with rank 4, degree 4, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(D, 64);[127X[104X
    [4X[28X<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XIsJordanDesignWithStrength(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignStrength(D);[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  
  [1X4.4-6 [33X[0;0YSchemes and Tight Designs[133X[101X
  
  [33X[1;0Y[29X[2XIsRegularSchemeJordanDesign[102X [32X filter[133X
  [33X[1;0Y[29X[2XIsAssociationSchemeJordanDesign[102X [32X filter[133X
  [33X[1;0Y[29X[2XIsTightJordanDesign[102X [32X filter[133X
  
  [33X[0;0YThese  filters  identify  various special categories of designs that satisfy
  [2XIsJordanDesignWithStrength[102X  ([14X4.4-5[114X).  In  what follows recall that [23Xt[123X denotes
  the strength of the design and [23Xs[123X denotes the cardinality of the angle set [23XA[123X.
  The definitions below are provided in [Hog92].[133X
  
  [33X[0;0YA  design  admits  a  [13Xregular  scheme[113X  when  [23Xt  \ge  s  -  1  [123X.  The  filter
  [10XIsRegularSchemeJordanDesign[110X  returns  true  when  both [23Xt[123X and [23Xs[123X are known and
  satisfy the regular scheme inequality given above.[133X
  
  [33X[0;0YA  design  admits  an  [13Xassociation  scheme[113X  when  [23Xt  \ge 2s - 2 [123X. The filter
  [10XIsAssociationSchemeJordanDesign[110X returns true when both [23Xt[123X and [23Xs[123X are known and
  satisfy the association scheme inequality given above.[133X
  
  [33X[0;0YFinally, a design is [13Xtight[113X when it satisfies [23Xt = 2s - 1[123X for [23X0[123X in [23XA[123X or [23Xt = 2s[123X
  otherwise.  The filter [10XIsTightJordanDesign[110X returns true when the appropriate
  equality is satisfied for a design.[133X
  
  
  [1X4.5 [33X[0;0YDesigns Admitting a Regular Scheme[133X[101X
  
  [1X4.5-1 JordanDesignSubdegrees[101X
  
  [33X[1;0Y[29X[2XJordanDesignSubdegrees[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YFor   a   design   [3XD[103X   with   cardinality   and  angle  set  that  satisfies
  [10XIsRegularSchemeJordanDesign[110X,  namely [23Xt \ge s - 1[123X, we can compute the regular
  subdegrees  as  described  in [Hog92, Theorem 3.2]. The subdegrees count the
  number  of  elements  forming each angle with some representative element in
  the design. So, in the example below, there are [23X27[123X elements forming an angle
  (inner product) of [23X1/9[123X with some representative design element.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4, 4, [1/3, 1/9]);; JordanDesignAddCardinality(D, 64);; D;[127X[104X
    [4X[28X<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignSubdegrees(D);[127X[104X
    [4X[28X[ 27, 36 ][128X[104X
  [4X[32X[104X
  
  
  [1X4.6 [33X[0;0YDesigns Admitting an Association Scheme[133X[101X
  
  [33X[0;0YWhen  a  design  satisfies  [23Xt  \ge 2s - 2[123X then it also admits an association
  scheme.  We  can use results given in [Hog92] to determine the parameters of
  the  corresponding  association  scheme.  For more details about association
  schemes see [CVL91] or [BBIT21].[133X
  
  [1X4.6-1 JordanDesignBoseMesnerAlgebra[101X
  
  [33X[1;0Y[29X[2XJordanDesignBoseMesnerAlgebra[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YFor  a  design that satisfies [10XIsAssociationSchemeJordanDesign[110X, we can define
  the  corresponding  Bose-Mesner  algebra  [BBIT21, pp. 53-57]. The canonical
  basis  for  this algebra corresponds to the adjacency matrices [23XA_i[123X, with the
  [10Xs+1[110X-th  basis vector corresponding to [23XA_0[123X. The adjacenty matrices themselves
  are  not  provided  and  the algebra is constructed from the known structure
  constants so that elements of this algebra satisfy [10XIsSCAlgebraObj[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4, 4, [1/3, 1/9]);; JordanDesignAddCardinality(D, 64);; D;[127X[104X
    [4X[28X<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XB := JordanDesignBoseMesnerAlgebra(D);[127X[104X
    [4X[28X<algebra of dimension 3 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XBasisVectors(CanonicalBasis(B));[127X[104X
    [4X[28X[ A1, A2, A3 ][128X[104X
    [4X[25Xgap>[125X [27XOne(B); IsSCAlgebraObj(last);[127X[104X
    [4X[28XA3[128X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.6-2 JordanDesignBoseMesnerIdempotentBasis[101X
  
  [33X[1;0Y[29X[2XJordanDesignBoseMesnerIdempotentBasis[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YFor  a  design  that  satisfies [10XIsAssociationSchemeJordanDesign[110X, we can also
  define  the  idempotent  basis  of  the  corresponding  Bose-Mesner  algebra
  [BBIT21, pp. 53-57].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4, 4, [1/3, 1/9]);; JordanDesignAddCardinality(D, 64);; D;[127X[104X
    [4X[28X<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27Xfor x in BasisVectors(JordanDesignBoseMesnerIdempotentBasis(D)) do Display(x); [127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[28X(-5/64)*A1+(3/64)*A2+(27/64)*A3[128X[104X
    [4X[28X(1/16)*A1+(-1/16)*A2+(9/16)*A3[128X[104X
    [4X[28X(1/64)*A1+(1/64)*A2+(1/64)*A3[128X[104X
    [4X[25Xgap>[125X [27XForAll(JordanDesignBoseMesnerIdempotentBasis(D), IsIdempotent);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.6-3 JordanDesignIntersectionNumbers[101X
  
  [33X[1;0Y[29X[2XJordanDesignIntersectionNumbers[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YThe     intersection     numbers     [23Xp^k_{i,j}[123X     are     given     by     [10X
  JordanDesignIntersectionNumbers([3XD[103X[10X)[k][i][j][110X.   These   intersection  numbers
  serve  as  the structure constants for the [10XJordanDesignBoseMesnerAlgebra([3XD[103X[10X)[110X.
  Namely,  [23XA_i  A_j  =  \sum_{k  =  1}^{s+1} p^{k}_{i,j} A_k[123X (see [BBIT21, pp.
  53-57]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4, 4, [1/3, 1/9]);; JordanDesignAddCardinality(D, 64);; D;[127X[104X
    [4X[28X<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XA := BasisVectors(Basis(JordanDesignBoseMesnerAlgebra(D)));;[127X[104X
    [4X[25Xgap>[125X [27Xp := JordanDesignIntersectionNumbers(D);;[127X[104X
    [4X[25Xgap>[125X [27XA[1]*A[2] = Sum(List([1..3]), k -> p[k][1][2]*A[k]);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.6-4 JordanDesignKreinNumbers[101X
  
  [33X[1;0Y[29X[2XJordanDesignKreinNumbers[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YThe       Krein       numbers       [23Xq^k_{i,j}[123X       are       given       by
  [10XJordanDesignKreinNumbers([3XD[103X[10X)[k][i][j][110X.   The   Krein  numbers  serve  as  the
  structure   constants   for   the  [10XJordanDesignBoseMesnerAlgebra([3XD[103X[10X)[110X  in  the
  idempotent basis given by [10XJordanDesignBoseMesnerIdempotentBasis([3XD[103X[10X)[110X using the
  Hadamard matrix product [23X\circ[123X. Namely, for idempotent basis [23XE_i[123X and Hadamard
  product  [23X\circ[123X,  we  have [23XE_i \circ E_j = \sum_{k = 1}^{s+1} q^{k}_{i,j} E_k[123X
  (see [BBIT21, pp. 53-57]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4, 4, [1/3, 1/9]);; JordanDesignAddCardinality(D, 64);; D;[127X[104X
    [4X[28X<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27Xq := JordanDesignKreinNumbers(D);; [127X[104X
    [4X[25Xgap>[125X [27XDisplay(q);[127X[104X
    [4X[28X[ [ [ 10, 16, 1 ], [ 16, 20, 0 ], [ 1, 0, 0 ] ], [128X[104X
    [4X[28X  [ [ 12, 15, 0 ], [ 15, 20, 1 ], [ 0, 1, 0 ] ], [128X[104X
    [4X[28X  [ [ 27, 0, 0 ], [ 0, 36, 0 ], [ 0, 0, 1 ] ] ][128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X4.6-5 JordanDesignFirstEigenmatrix[101X
  
  [33X[1;0Y[29X[2XJordanDesignFirstEigenmatrix[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YAs  describe  in  [BBIT21,  p.  58],  the first eigenmatrix of a Bose-Mesner
  algebra  [23XP_i(j)[123X  defines  the expansion of the adjacency matrix basis [23XA_i[123X in
  terms  of  the  idempotent  basis  [23XE_j[123X  as follows: [23XA_i = \sum_{j = 1}^{s+1}
  P_i(j)   E_j   [123X.   This   attribute   returns   the   component   [23XP_i(j)[123X  as
  [10XJordanDesignFirstEigenmatrix([3XD[103X[10X)[i][j][110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4,4,[1/3,1/9]);; JordanDesignAddCardinality(D, 64);; D;[127X[104X
    [4X[28X<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27Xa := Basis(JordanDesignBoseMesnerAlgebra(D));;[127X[104X
    [4X[25Xgap>[125X [27Xe := JordanDesignBoseMesnerIdempotentBasis(D);;[127X[104X
    [4X[25Xgap>[125X [27XForAll([1..3], i -> a[i] = Sum([1..3], j ->[127X[104X
    [4X[25X>[125X [27XJordanDesignFirstEigenmatrix(D)[i][j]*e[j]));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.6-6 JordanDesignSecondEigenmatrix[101X
  
  [33X[1;0Y[29X[2XJordanDesignSecondEigenmatrix[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YAs  describe  in  [BBIT21,  p.  58], the second eigenmatrix of a Bose-Mesner
  algebra [23XQ_i(j)[123X defines the expansion of the idempotent basis [23XE_i[123X in terms of
  the  adjacency  matrix  basis  [23XA_j[123X as follows: [23XE_i = (1/v)\sum_{j = 1}^{s+1}
  Q_i(j)   A_j   [123X.   This   attribute   returns   the   component   [23XQ_i(j)[123X  as
  [10XJordanDesignSecondEigenmatrix([3XD[103X[10X)[i][j][110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4,4,[1/3,1/9]);; JordanDesignAddCardinality(D, 64);; D;[127X[104X
    [4X[28X<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27Xa := Basis(JordanDesignBoseMesnerAlgebra(D));;[127X[104X
    [4X[25Xgap>[125X [27Xe := JordanDesignBoseMesnerIdempotentBasis(D);;[127X[104X
    [4X[25Xgap>[125X [27XForAll([1..3], i -> e[i]*JordanDesignCardinality(D) =[127X[104X
    [4X[25X>[125X [27XSum([1..3], j -> JordanDesignSecondEigenmatrix(D)[i][j]*a[j]));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignFirstEigenmatrix(D) = Inverse(JordanDesignSecondEigenmatrix(D))[127X[104X
    [4X[25X>[125X [27X*JordanDesignCardinality(D);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.6-7 JordanDesignMultiplicities[101X
  
  [33X[1;0Y[29X[2XJordanDesignMultiplicities[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YAs  describe in [BBIT21, pp. 58-59], the design multiplicy [23Xm_i[123X is defined as
  the  dimension of the space that idempotent matrix [23XE_i[123X projects onto, or [23Xm_i
  = \mathrm{Tr}(E_i)[123X. We also have [23Xm_i = Q_i(s+1)[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4,4,[1/3,1/9]);; JordanDesignAddCardinality(D, 64);; D;[127X[104X
    [4X[28X<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignMultiplicities(D);[127X[104X
    [4X[28X[ 27, 36, 1 ][128X[104X
  [4X[32X[104X
  
  [1X4.6-8 DesignValencies[101X
  
  [33X[1;0Y[29X[2XDesignValencies[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YAs  describe  in  [BBIT21, pp. 55, 59], the design valency [23Xk_i[123X is defined as
  the  fixed  number  of [23Xi[123X-associates of any element in the association scheme
  (also known as the subdegree). We also have [23Xk_i = P_i(s+1)[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4,4,[1/3,1/9]);; JordanDesignAddCardinality(D, 64);; D;[127X[104X
    [4X[28X<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XDesignValencies(D);[127X[104X
    [4X[28X[ 27, 36, 1 ][128X[104X
  [4X[32X[104X
  
  [1X4.6-9 JordanDesignReducedAdjacencyMatrices[101X
  
  [33X[1;0Y[29X[2XJordanDesignReducedAdjacencyMatrices[102X( [3XD[103X ) [32X attribute[133X
  
  [33X[0;0YAs  defined in [CVL91, p. 201], the reduced adjacency matrices multiply with
  the  same  structure constants as the adjacency matrices, which allows for a
  simpler  construction  of  an algebra isomorphic to the Bose-Mesner algebra.
  The  matrices  [10XJordanDesignReducedAdjacencyMatrices([3XD[103X[10X)[110X are used to construct
  [10XJordanDesignBoseMesnerAlgebra([3XD[103X[10X)[110X.[133X
  
  
  [1X4.7 [33X[0;0YExamples[133X[101X
  
  [33X[0;0YThis  section  provides a number of known examples that can be studied using
  the [5XALCO[105X package. The following tight projective t-designs are identified in
  [Hog82, Examples 1-11].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(2, 1, [0,1/2]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 3-design with rank 2, degree 1, cardinality 4, and angle set [128X[104X
    [4X[28X[ 0, 1/2 ]> [128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(2, 2, [0,1/2]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 3-design with rank 2, degree 2, cardinality 6, and angle set[128X[104X
    [4X[28X[ 0, 1/2 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(2, 4, [0,1/2]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 3-design with rank 2, degree 4, cardinality 10, and angle set[128X[104X
    [4X[28X[ 0, 1/2 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(2, 8, [0,1/2]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 3-design with rank 2, degree 8, cardinality 18, and angle set[128X[104X
    [4X[28X[ 0, 1/2 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(3, 2, [1/4]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 2-design with rank 3, degree 2, cardinality 9, and angle set [ 1/4 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(4, 2, [0,1/3]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 3-design with rank 4, degree 2, cardinality 40, and angle set[128X[104X
    [4X[28X[ 0, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(6, 2, [0,1/4]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 3-design with rank 6, degree 2, cardinality 126, and angle set[128X[104X
    [4X[28X[ 0, 1/4 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(8, 2, [1/9]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 2-design with rank 8, degree 2, cardinality 64, and angle set [ 1/9 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(5, 4, [0,1/4]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 3-design with rank 5, degree 4, cardinality 165, and angle set[128X[104X
    [4X[28X[ 0, 1/4 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(3, 8, [0,1/4,1/2]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 5-design with rank 3, degree 8, cardinality 819, and angle set[128X[104X
    [4X[28X[ 0, 1/4, 1/2 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(24, 1, [0,1/16,1/4]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 5-design with rank 24, degree 1, cardinality 98280, and angle set[128X[104X
    [4X[28X[ 0, 1/16, 1/4 ]>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAn additional icosahedron projective example is identified in [Lyu09].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(2, 2, [ 0, (5-Sqrt(5))/10, (5+Sqrt(5))/10 ]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 5-design with rank 2, degree 2, cardinality 12, and angle set[128X[104X
    [4X[28X[ 0, -3/5*E(5)-2/5*E(5)^2-2/5*E(5)^3-3/5*E(5)^4,[128X[104X
    [4X[28X  -2/5*E(5)-3/5*E(5)^2-3/5*E(5)^3-2/5*E(5)^4 ]>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  Leech  lattice  short  vector  design and several other tight spherical
  designs are given below:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(2, 23, [ 0, 1/4, 3/8, 1/2, 5/8, 3/4 ]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 11-design with rank 2, degree 23, cardinality 196560, and angle set[128X[104X
    [4X[28X[ 0, 1/4, 3/8, 1/2, 5/8, 3/4 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(2, 5, [ 1/4, 5/8 ]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 4-design with rank 2, degree 5, cardinality 27, and angle set[128X[104X
    [4X[28X[ 1/4, 5/8 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(2, 6, [0,1/3,2/3]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 5-design with rank 2, degree 6, cardinality 56, and angle set[128X[104X
    [4X[28X[ 0, 1/3, 2/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(2, 21, [3/8, 7/12]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 4-design with rank 2, degree 21, cardinality 275, and angle set[128X[104X
    [4X[28X[ 3/8, 7/12 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(2, 22, [0,2/5,3/5]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 5-design with rank 2, degree 22, cardinality 552, and angle set[128X[104X
    [4X[28X[ 0, 2/5, 3/5 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(2, 7, [0,1/4,1/2,3/4]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 7-design with rank 2, degree 7, cardinality 240, and angle set[128X[104X
    [4X[28X[ 0, 1/4, 1/2, 3/4 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(2, 22, [0,1/3,1/2,2/3]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<Tight 7-design with rank 2, degree 22, cardinality 4600, and angle set[128X[104X
    [4X[28X[ 0, 1/3, 1/2, 2/3 ]>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSome  projective  designs  meeting  the  special  bound are given in [Hog82,
  Examples 1-11]:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(4, 4, [0,1/4,1/2]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<3-design with rank 4, degree 4, cardinality 180, and angle set[128X[104X
    [4X[28X[ 0, 1/4, 1/2 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(3, 2, [0,1/3]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<2-design with rank 3, degree 2, cardinality 12, and angle set [ 0, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(5, 2, [0,1/4]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<2-design with rank 5, degree 2, cardinality 45, and angle set [ 0, 1/4 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(9, 2, [0,1/9]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<2-design with rank 9, degree 2, cardinality 90, and angle set [ 0, 1/9 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(28, 2, [0,1/16]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<2-design with rank 28, degree 2, cardinality 4060, and angle set [ 0, 1/16 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(4, 4, [0,1/4]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<2-design with rank 4, degree 4, cardinality 36, and angle set [ 0, 1/4 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(4, 4, [1/9,1/3]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<2-design with rank 4, degree 4, cardinality 64, and angle set [ 1/9, 1/3 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(16, 1, [0,1/9]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<2-design with rank 16, degree 1, cardinality 256, and angle set [ 0, 1/9 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(4, 2, [0,1/4,1/2]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<3-design with rank 4, degree 2, cardinality 60, and angle set[128X[104X
    [4X[28X[ 0, 1/4, 1/2 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(16, 1, [0,1/16,1/4]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<3-design with rank 16, degree 1, cardinality 2160, and angle set[128X[104X
    [4X[28X[ 0, 1/16, 1/4 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(3, 4, [0,1/4,1/2]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<3-design with rank 3, degree 4, cardinality 63, and angle set[128X[104X
    [4X[28X[ 0, 1/4, 1/2 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(3, 4, [0,1/4,1/2,(3+Sqrt(5))/8, (3-Sqrt(5))/8]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<5-design with rank 3, degree 4, cardinality 315, and angle set[128X[104X
    [4X[28X[ 0, 1/4, 1/2, -1/2*E(5)-1/4*E(5)^2-1/4*E(5)^3-1/2*E(5)^4,[128X[104X
    [4X[28X  -1/4*E(5)-1/2*E(5)^2-1/2*E(5)^3-1/4*E(5)^4 ]>[128X[104X
    [4X[25Xgap>[125X [27XJordanDesignByAngleSet(12, 2, [0,1/3,1/4,1/12]);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<5-design with rank 12, degree 2, cardinality 32760, and angle set[128X[104X
    [4X[28X[ 0, 1/12, 1/4, 1/3 ]>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTwo important designs related to the [23XH_4[123X Weyl group are as follows:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA := [ 0, 1/4, 1/2, 3/4, (5-Sqrt(5))/8, (5+Sqrt(5))/8,[127X[104X
    [4X[25X>[125X [27X(3-Sqrt(5))/8, (3+Sqrt(5))/8 ];;[127X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(2, 3, A);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<11-design with rank 2, degree 3, cardinality 120, and angle set[128X[104X
    [4X[28X[ 0, 1/4, 1/2, 3/4, -3/4*E(5)-1/2*E(5)^2-1/2*E(5)^3-3/4*E(5)^4,[128X[104X
    [4X[28X  -1/2*E(5)-3/4*E(5)^2-3/4*E(5)^3-1/2*E(5)^4,[128X[104X
    [4X[28X  -1/2*E(5)-1/4*E(5)^2-1/4*E(5)^3-1/2*E(5)^4,[128X[104X
    [4X[28X  -1/4*E(5)-1/2*E(5)^2-1/2*E(5)^3-1/4*E(5)^4 ]>[128X[104X
    [4X[25Xgap>[125X [27XA := [ 0, 1/4, (3-Sqrt(5))/8, (3+Sqrt(5))/8 ];;[127X[104X
    [4X[25Xgap>[125X [27XD := JordanDesignByAngleSet(4, 1, A);;[127X[104X
    [4X[25Xgap>[125X [27XJordanDesignAddCardinality(last, JordanDesignSpecialBound(last));[127X[104X
    [4X[28X<5-design with rank 4, degree 1, cardinality 60, and angle set[128X[104X
    [4X[28X[ 0, 1/4, -1/2*E(5)-1/4*E(5)^2-1/4*E(5)^3-1/2*E(5)^4,[128X[104X
    [4X[28X  -1/4*E(5)-1/2*E(5)^2-1/2*E(5)^3-1/4*E(5)^4 ]>[128X[104X
  [4X[32X[104X
  
