Coding Theory and Design Theory pp 35-50 | Cite as

# Families of Codes with Few Distinct Weights from Singular and Non-Singular Hermitian Varieties and Quadrics in Projective Geometries and Hadamard Difference Sets and Designs Associated with Two-Weight Codes

## Summary

In this paper, we present several doubly infinite families of linear projective codes with two-, three- and five distinct non-zero Hamming weights together with the frequency distributions of their weights.

The codes have been defined as linear spaces of coordinate vectors of points on certain projective sets described in terms of Hermitian and quadratic forms - nondegenerate and singular - in projective spaces. The weight-distributions have been derived by considering the geometry of intersections of projective sets by hyperplanes in relevant projective spaces. Results from Bose and Chakravarti (1966) and Chakravarti (1971) on the Hermitian geometry and Bose (1964), Primrose (1951) and Ray-Chaudhuri (1959, 1962) have been used in the enumeration of weights and their frequencies.

The paper has been organized as follows. Preliminary definitions, concepts and results on Hermitian geometry [from Bose and Chakravarti (1966) and Chakravarti (1971)] are given in Section 1.

Two families of two-weight codes \(
e\left( {V_{N - 1} } \right)
\) and \(
e\left( {\bar V_{N - 1} } \right)
\) over *GF*(*s* ^{2}) and associated families \(
e'\left( {V_{N - 1} } \right)
\) and \(
e'\left( {\bar V_{N - 1} } \right)
\) over *GF(s)* together with their weight-distributions are given in Section 2. Here \(V_{N-1}\) denotes a non-degenerate Hermitian variety in *PG*(*N*,*s* ^{2}) and \(\bar V_{N-1}\) is its complement and a code \(
e\left( S \right)
\) is defined as the linear space of the coordinate vectors of the points in the projective set *S*. These codes have been otherwise obtained by Wolfmann (1975, 1977) from quadrics and by Calderbank and Kantor (1986) from the rank three representation of unitary groups. However, this latter paper was not available to the author while he presented his results at Marseille (1986).

The eigenvalues of the adjacency matrix *A* = *B* _{2} - *B* _{1} (*B* _{i} is the incidence matrix of the ith associates *i* = 1,2) of the strongly regular graph (two-class association scheme) on s^{2(N+1)} vertices defined by the two-weight code \(
e'\left( {V_{N - 1} } \right)
\) over *GF(s)* and the (*p* _{jk} ^{i} ) parameters of the two-class association scheme are given in Section 3.

In section 4, we show that for *s* = 2, *B* _{2} (the association matrix of the second associates) of the two-class association scheme of Section 3, is the incidence matrix a symmetric BIB design with parameters \(\upsilon = 2^{2(N+1)}, k = 2^{2(N+1)} + (-2)^N, \lambda = 2^{2N} + (-2)^N\) and 2*B* _{2} - *J* is a Hadamard matrix of order 2^{2(N+1)}. Similarly, *I* + *B* _{1} is the incidence matrix of a symmetric BIB design with parameters \(\upsilon = 2^{2(N+1)}, k = 2^{2(N+1)} - (-2)^N\). Further, it is shown that the 2^{2N+1} + (-2)^{ N } codewords each of weight (2^{2N } - (-2)^{ N }), which are non-adjacent to the null codeword form a Hadamard difference set (Menon 1960, Mann 1965) with parameters υ = 2^{2N+2}, *k* = 2^{2N+1} + (-2)^{ N }, λ = 2^{2N } + (-2)^{ N } and the (2^{2N+1} - (-2)^{ N } - 1) codewords each of weight 2^{2N } together with the null codeword form a Hadamard difference set with parameters υ = 2^{2N+2}, *k* = 2^{2N+1} - (-2)^{ N }, λ = 2^{2N } - (-2)^{ N }, for integer *N*. These difference sets also appear in Wolfmann (1977) and Calderbank and Kantor (1986). But our presentation in terms association matrices is of special interest to statisticians.

In Section 5, a family of five-weight linear codes and the associated weight-distributions are derived. A code here is defined as the linear span of a projective set which is the intersection of a non-degenerate Hermitian variety and the complement of one of the secant hyperplanes. These codes are believed to be new.

In Sections 6 and 7, we consider codes which are linear spans of projective sets defined in terms of degenerate Hermitian and quadratic forms in projective spaces. The motivation here is to explore how the code parameters behave when the basic projective set is not purely a subspace nor a non-degenerate Hermitian or quadric variety but an amalgam of the two, which still admits a geometric description (and algebraic equations).

In section 6, the basic projective set is a degenerate Hermitian variety \(V^o_{N-2}\) which is the intersection of a non-degenerate Hermitian variety \(V_{N-1}\) in *PG*(*N*, *s* ^{2}) with one of its tangent hyperplanes. The code \(
e\left( {V_{N - 1}^ \circ } \right)
\) which is the linear space generated by the coordinate vectors of the points of \(V^o_{N-2}\), is shown to be a triweight code. Its weight-distribution as a code over *GF*(*s* ^{2}) as well as that of its sister code over *GF(s)* are given. This family seems to be new.

In section 7, a degenerate quadric \(Q^o_{N-1}\) which is the intersection of a nondegenerate quadric *Q* _{n} in *PG*(*N*, *s*) with one of its tangent hyperplanes, is taken as the basic projective set. The code \(
e\left( {Q_{N - 1}^ \circ } \right)
\) which is the linear space of the coordinate vectors of the points of \((Q^o_{N-1})\)is shown to be a tri-weight code both for odd and even *N*. The frequency distributions of the weights are given for both odd and even *N*. For odd *N*, both the cases elliptic and hyperbolic have been considered. These families supplement whose obtained by Wolfmann (1975) from non-degenerate quadrics, and these codes for odd *N*, are believed to be new. For even *N* and *s* = 2, this code was given by Dowling (1969). This is not a cyclic code, but he showed that this can be made cyclic by adding all permutations of the codewords and 2(2^{2t} - 1) other codewords. The weight-distribution of the code given in our Table 7.1, corresponds to Games’s (1986) table for *N* = 2*t* - 1, *r* = 1, *q* = *s*. Games (1986) calculated the sizes and their respective multiplicities of intersections by hyperplanes of a degenerate quadric (cone) of order *r* in *PG*(*N*, *q*), for *N* - *r* even.

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## References

- [1]Bose, R.C., Lecture Notes on
*Combinatorial Problems of Experimental Design*, Department of Statistics, University of North Carolina at Chapel Hill (1962).Google Scholar - [2]Bose, R.C.,
*On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements*, Calcutta Math. Soc. Golden Jubilee Comm., Part II, 1958–59 (1963), pp. 341–356.Google Scholar - [3]Bose, R.C.,
*Self-conjugate tetrahedra with respect to the Hermitian variety*x^{3}_{0}+x^{3}_{1}+x^{3}_{2}+x^{3}_{3}= 0*in PG*(3,2^{2})*and a representation of PG*(3,3), Proc. Symp. on Pure Math., 19 (Amer. Math. Soc. Providence, RI) (1971), pp. 27–37.Google Scholar - [4]Bose, R.C. and Chakravarti, I.M.,
*Hermitian varieties in a finite projective space PG(N, q*^{2}), Canad. J. Math., 18 (1966), pp. 1161–1182.MathSciNetzbMATHCrossRefGoogle Scholar - [5]Bose R.C. and Mesner, D.M.,
*On linear associative algebras corresponding to association schemes of partially balanced designs*, Ann. Math. Statist., 30 (1959), pp. 21–38.MathSciNetzbMATHCrossRefGoogle Scholar - [6]Brouwer, A.E.,
*Some new two-weight codes and strongly regular graphs*, Discrete Appl. Math., 10 (1985), pp. 111–114.MathSciNetzbMATHCrossRefGoogle Scholar - [7]Calderbank, R. and Kantor, W.M.,
*The geometry of two-weight codes*, Bull. London Math. Soc., 18 (1986), pp. 97–122.MathSciNetzbMATHCrossRefGoogle Scholar - [8]Chakravarti, I.M.,
*Some properties and applications of Hermitian varieties in PG(N,q*^{2}) in*the construction of strongly regular graphs (two-class association schemes) and block designs*, Journal of Comb. Theory, Series B, 11(3) (1971), pp. 268–283.MathSciNetzbMATHCrossRefGoogle Scholar - [9]Chakravarti, I.M.,
*The generalized Goppa codes and related discrete designs from Hermitian varieties*, Institute of Statistics Mimeo Series 1713. Department of Statistics, University of North Carolina at Chapel Hill.Google Scholar - [10]Delsarte, P.,
*Weights of linear codes and strongly regular normed spaces*, Discrete Math., 3 (1972), pp. 47–64.MathSciNetzbMATHCrossRefGoogle Scholar - [11]Delsarte, P., An
*algebraic approach to the association schemes of coding theory*, Philips. Res. Rep. Suppl., 19 (1973).Google Scholar - [12]
- [13]Dickson, L.E.,
*Linear Groups with an Exposition of the Galois Field Theory*, Teubner, Dover Publications Inc., New York, (1901, 1958).zbMATHGoogle Scholar - [14]Dieudonné, J.,
*La Géométrie des Groupes Classiques*, Springer-Verlag, Berlin, Troisième Edition, 1971.zbMATHGoogle Scholar - [15]Dowling, T.A.,
*A class of tri-weight codes*, Institute of Statistics Mimeo Series No. 600.3. University of North Carolina at Chapel Hill, Department of Statistics (1969).Google Scholar - [16]Games, R.A.,
*The geometry of quadrics and correlations of sequences*, IEEE Trans. Inf. Th., IT-32 (1986), pp. 423–426.MathSciNetCrossRefGoogle Scholar - [17]Heft, S.M.,
*Spreads in Projective Geometry and Associated Designs*, Ph.D. dissertation submitted to the University of North Carolina, Dept, of Statistics, Chapel Hill (1971).Google Scholar - [18]Higman, D.J. and McLaughlin, J.E.,
*Rank 3 subgroups of finite symplectic and unitary groups*, J. Reine Angew. Math., 218 (1965), pp. 174–189.MathSciNetzbMATHGoogle Scholar - [19]
- [20]Jordan C.,
*Traité des Substitutions et des Equations Algébriques*, Gauthier-Villars, Paris, 1870.Google Scholar - [21]Mac Williams, F.J. and Sloane, N.J.A.,
*The Theory of Error-Correcting Codes*, North Holland, 1977.zbMATHGoogle Scholar - [22]MacWilliams, F.J., Odlyzko, A.M., Sloane, N.J.A. and Ward, H.N.,
*Self-dual codes over GF(*4), J. Comb. Th., A25 (1978), pp. 288–318.MathSciNetGoogle Scholar - [23]
- [24]Menon, P.K.,
*Difference sets in Abelian groups*, Proc. Amer. Math. Soc., 11 (1960), pp. 368–376.MathSciNetzbMATHCrossRefGoogle Scholar - [25]Mesner, D.M., A
*new family of partially balanced incomplete block designs with some latin square design properties*, Ann. Math. Statist., 38 (1967), pp. 571–581.MathSciNetzbMATHCrossRefGoogle Scholar - [26]Primrose, E.J.F.,
*Quadrics in finite geometries*, Proc. Comb. Phil. Soc., 47 (1951), pp. 299–304.MathSciNetzbMATHCrossRefGoogle Scholar - [27]Ray-Chaudhuri, D.K.,
*On the application of the geometry of quadrics to the construction of partially balanced incomplete block designs and error correcting codes*, Ph.D. dissertation submitted to the University of North Carolina at Chapel Hill (1959).Google Scholar - [28]Ray-Chaudhuri, D.K.,
*Some results on quadrics in finite projective geometry based on Galois fields*, Canad. J. Math., 14 (1962), pp. 129–138.MathSciNetzbMATHCrossRefGoogle Scholar - [29]Segre B.,
*Forme e geometrie hermitiane, con particolare riguardo al caso finito*, Ann. Math. Pure Appl., 70 1 (1965), p. 202.MathSciNetGoogle Scholar - [30]Segre, B.,
*Introduction to Galois Geometries, Atti della Acc. Nazionale dei*, Lincei, Roma, 8(5) (1967), pp. 137–236.MathSciNetGoogle Scholar - [31]Wolfmann, J.,
*Codes projectifs à deux ou trois poids associés aux hyperquadriques d’une géométrie finie*, Discrete Mathematics, 13 (1975), pp. 185–211.MathSciNetzbMATHCrossRefGoogle Scholar - [32]Wolfmann, J.,
*Codes projectifs à deux poids, “caps” complets et ensembles de differences*, J. Combin. Theory, 23A (1977), pp. 208–222.MathSciNetCrossRefGoogle Scholar