Five-weight codes from three-valued correlation of M-sequences

Abstract

<p style='text-indent:20px;'>In this paper, for each of six families of three-valued <inline-formula><tex-math id="M1">\begin{document}$ m $\end{document}</tex-math></inline-formula>-sequence correlation, we construct an infinite family of five-weight codes from trace codes over the ring <inline-formula><tex-math id="M2">\begin{document}$ R = \mathbb{F}_2+u\mathbb{F}_2 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ u^2 = 0. $\end{document}</tex-math></inline-formula> The trace codes have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Their support structure is determined. An application to secret sharing schemes is given. The parameters of the binary image are <inline-formula><tex-math id="M4">\begin{document}$ [2^{m+1}(2^m-1),4m,2^{m}(2^m-2^r)] $\end{document}</tex-math></inline-formula> for some explicit <inline-formula><tex-math id="M5">\begin{document}$ r. $\end{document}</tex-math></inline-formula></p>

Lee weight distributions of trace codes over F_q+uF_q from irreducible cyclic codes

“…In a similar way for a trace code C L , defined through (1), let A j (0 ≤ j ≤ 2n) be the number of codewords with Lee weight j in the linear code C L of length n over R, then the Lee weight enumerator of C L , Lee C L (z), is defined by Lee C L (z) = 2n j=0 A j z j . Through different choices of the defining set L, and particularly when F q is either the binary field (F 2 ) or a prime field (F p ), several optimal or nearly optimal codes from trace codes of the form C L where recently found ( [4,5,8,9,[11][12][13][14][15][16]). Particularly, the construction of two or few-weight codes from trace codes over the ring F q + uF q , was recently presented in [4].…”

“…For such construction, the defining sets for the trace codes are given in terms of cyclotomic classes, and for some of these classes, it is shown that it is possible to obtain the Lee weight distributions of the corresponding trace codes. Motivated by this construction, and by the p-ary semiprimitive irreducible cyclic codes over a prime field F p , the Lee weight distributions of an infinite family of p-ary three-weight codes from trace codes over the ring F p + uF p , was recently constructed in [11].…”

“…One of the purposes of this work is to show that there exists a direct identity between Lee weight enumerators of the trace codes constructed by [4], [8], [9] and [11], and the Hamming weight enumerators of the irreducible cyclic codes. In other words, we are going to prove that the Lee weight distribution problem for the trace codes constructed following [4], [8], [9] or [11], is equivalent to the standard Hamming weight distribution problem for the irreducible cyclic codes.…”

“…One of the purposes of this work is to show that there exists a direct identity between Lee weight enumerators of the trace codes constructed by [4], [8], [9] and [11], and the Hamming weight enumerators of the irreducible cyclic codes. In other words, we are going to prove that the Lee weight distribution problem for the trace codes constructed following [4], [8], [9] or [11], is equivalent to the standard Hamming weight distribution problem for the irreducible cyclic codes. With this equivalence in mind, we then follow the open problem suggested in the Conclusion of [11], and determine the Lee weight distribution of trace codes of the form C L in terms of other class of irreducible cyclic codes, with known or well-understood weight distribution.…”

“…Motivated by this construction, and by the p-ary semiprimitive irreducible cyclic codes over a prime field Fp, the Lee weight distributions of an infinite family of p-ary three-weight codes from trace codes over the ring Fp + uFp, was recentely found in [11]. In this work, we prove that the Lee weight distribution problem for the trace codes constructed in accordance with either [4] or [11], is equivalent to the weight distribution problem for the irreducible cyclic codes. With this equivalence in mind, and by using the already known weight distributions of an infinite family of irreducible cyclic codes (semiprimitive and not semiprimitive), we follow the open problem suggested in the Conclusion of [11] to determine the Lee weight distribution of an infinite family of trace codes over the ring Fq + uFq, that includes the infinite family found in [11].…”