Optimal equi-difference conflict-avoiding codes of odd length and weight three

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Abstract

A conflict-avoiding code (CAC) is known as a protocol sequence for transmitting data packets over a collision channel without feedback. The study of CACs has been focused on determining the size of an optimal code, i.e., the maximum size of a code, and in the past few years it has been settled by several researchers for even length and weight 3 together with constructions. As for odd length, a necessary and sufficient condition for the existence of a ‘tight equi-difference’ CAC of weight 3 can be found in Momihara (2007), but the condition is fairly complex and thus only a few explicit series of code lengths are known. Recently, Fu et al. (2013) restated the condition given by Momihara (2007) in a different way, which requires to examine the multiplicative suborder of 2 modulo p for each prime factor p of m. Meanwhile, Ma et al. (2013) presented constructions of an optimal equi-difference CAC and an optimal tight CAC of odd prime length p and weight 3, and formulated the sizes of such optimal codes. However, for their formulae to have practical meaning, the number of cosets of (2)p(2)p still needs to be determined, where (2)p is the multiplicative subgroup of Zp with generator 2. Moreover, their construction of an optimal tight CAC imposes a certain condition. This implies that even restricting ourselves to odd prime length, to provide a series of odd code length for which the maximum size of a CAC of weight 3 can be determined is a demanding problem.

In this article, we will give some explicit series of tight/optimal equi-difference CACs of odd length and weight 3 by revisiting some properties of multiplicative order of a unit in the ring of residues modulo m and cyclotomic polynomials.

MSC

05B30
05B99
94B25
94B65

Keywords

CAC
Equi-difference conflict-avoiding code
Optimal code
Tight code
Multiplicative order
Cyclotomic polynomial

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