New Classes of p-Ary Few Weight Codes

We generalise Sidel’nikov’s theorem from binary codes to $q$-ary codes for $q>2$. Denoting by $A(z)$ the cumulative distribution function attached to the weight distribution of the code and by $\unicode[STIX]{x1D6F7}(z)$ the standard normal distribution function, we show that $|A(z)-\unicode[STIX]{x1D6F7}(z)|$ is bounded above by a term which tends to $0$ when the code length tends to infinity.

show abstract

“…We use this notation throughout the paper. Further details on q-ary linear codes can be found in [6][7][8][9][10].…”

Section: Preliminariesmentioning

confidence: 99%

On the Generalisation of Sidel’nikov’s Theorem to -Ary Linear Codes

Wei

Wu²,

Wang³

2019

Bull. Aust. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Vega

2023

J. Algebra Comb. Discrete Appl.

View full text Add to dashboard Cite

The construction of two or few-weight codes from trace codes over the ring Fq + uFq, where u2 = 0, 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 Fp, the Lee weight distributions of an infinite family of p-ary three-weight codes from trace codes over the ring Fq + uFq, 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 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].

show abstract

“…So it has provoked tremendous interest in determining the complete weight enumerators and weight enumerators for linear codes. The authors in [1,2,3,8,14,18,19,23,24,25,26,27,28,29,31,32,33,34] dealt with this topic for different kinds of linear codes over finite fields and rings. The motivation of such research is that the weight distribution of a code allows the computation of the error probability of error detection and correction with respect to some algorithms.…”

Section: Introductionmentioning

confidence: 99%