Macaulay, Lazard and the Syndrome Variety

Abstract

In this paper we consider the four syndrom varieties Z × e , i.e. the set of all error locations corresponding to errors of weight w, 0 ≤ w ≤ 2, Z × ns , the set of all non spurious error locations corresponding to errors of weight w, 0 ≤ w ≤ 2, Z × + , the set of all non-spurious error locations corresponding to errors of weight w, 1 ≤ w ≤ 2, Z × 2 , the set of all non-spurious error locations corresponding to errors of weight w = 2, associated to an up-to-two errors correcting binary cyclic codes. Denoting J * := I(Z * ), the ideal of these syndrome varieties, N * := N(J * ) the Gröbner escalier of J * w.r.t. the lex ordering with x1 < x2 < z1 < z2, Φ * : Z * → N * a Cerlienco-Mureddu correspondence, and G * a minimal Groebner basis of the ideal J * , the aim of the paper is, assuming to know the structure of the order ideal N2 and a Cerlienco Mureddu Correspondence to deduce with elementary arguments N * , G * and Φ * for * ∈ {e, ns, +}.The tools are Macaulay's trick and Lazard's formulation of Cerlienco-Mureddu correspondence. * The author expresses her heartful thanks to Teo Mora for the many fruitful discussions and suggestions on this topic. 1 The recent mood, not depending on the network technology of the Sixties, and thus not needing the key equation, prefers consider the plain error locator polynomial µ j=1 (x − α ℓ j ).

HELP: a sparse error locator polynomial for BCH codes

“…Indeed, if m 3 > 2 then the term z 2 1 would be in the escalier, contradicting Theorem 9. Similarly (see also [11]), m 1 = n, m 2 = n+1 2 and m 4 = 1. By Theorem 11, then, the polynomial f in the minimal Groebner basis, with T( f ) = z 2 1 can be decomposed in…”

“…For an extensive study of the escalier associated to the syndrome varieties, both for n = 2 m − 1 and for the case n | 2 m − 1, see [11]. The paper, indeed, examines all the possible cases and computes all the related escaliers.…”

“…These stripes are posed in r columns depending on the Zech logarithm's structure [12]. We are studying this more general case in the works in progress [10,11].…”