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 ⁱ_jk ) 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 2B 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 = 2t - 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.