Higher Weight Spectra of Veronese Codes

Abstract

We study q-ary linear codes C obtained from Veronese surfaces over finite fields. We show how one can find the higher weight spectra of these codes, or equivalently, the weight distribution of all extension codes of C over all field extensions of Fq. Our methods will be a study of the Stanley-Reisner rings of a series of matroids associated to each code C.

Weight Spectra of Gabidulin Rank-metric Codes and Betti Numbers

“…• Theorem 61 is an analogue of [13,Corollary 17], which applies to Hamming codes and associated matroids. In both [13] and [14] one found all the φ (l) j , and using this corollary, one found all the weight spectra, i.e. all the A (j) s for two kinds of Veronese codes.…”

Möbius and coboundary polynomials for matroids

“…These quantities appear naturally when computing the generalized weight polynomials of the higher weight spectra of a linear code. Namely, from [10], the knowledge of all the generated weight polynomials, or all of the higher weight spectra, or of all the φ j for the associated matroid and (all of) its elongations are equivalent. Moreover, from [4] it is known that the knowledge of each of these three information pieces is equivalent to knowing the Tutte polynomials, and therefore the two-variable coboundary polymomials of the associated matroid and its dual.…”

“…Example 33 In [10], one describes so-called Veronese codes for all prime powers q. For q = 5 such a code is a linear [31, 6] 5 -code, and one describes its generalized Hamming weights and higher weight spectra in detail.…”

Free Resolutions and Generalized Hamming Weights of binary linear codes

“…In their seminal paper [17], Johnsen and Verdure showed how the GHWs of a linear code could be computed from a free resolution of a monomial ideal associated to the set of codewords of minimal support of the code provided we know this last set. That paper has produced a great avenue of research, see for example [9,11,18,19,20,21,22,23,24].…”