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

机译：奇数长度和权重3的最佳等差避免冲突代码

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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)_pU(2)_p still needs to be determined, where (2)_p is the multiplicative subgroup of Z_p~* 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.
机译：避免冲突码（CAC）被称为协议序列，用于通过冲突信道在没有反馈的情况下传输数据包。对CAC的研究一直专注于确定最佳代码的大小，即代码的最大大小，并且在过去几年中，几位研究人员已针对均匀长度和权重3及其结构确定了该代码。关于奇数长度，可以在Momihara（2007）中找到权重为3的“紧密等差” CAC的存在的充要条件，但条件相当复杂，因此只有几个明确的代码长度系列众所周知。最近，傅等人。 （2013年）以不同的方式重申了Momihara（2007年）给出的条件，该条件要求检查m的每个素数p的2模p的乘法子阶。同时，马等。 （2013年）提出了奇等长p和权重3的最佳等差CAC和最佳紧密CAC的构造，并制定了此类最佳代码的大小。但是，为了使它们的公式具有实际意义，仍需要确定-（2）_pU（2）_p的陪集数，其中（2）_p是Z_p〜*与生成器2的乘子组。最佳紧密CAC的构造会施加一定条件。这意味着，即使将自己限制在奇数素数长度上，以提供一系列奇数码长，针对该奇数码长，可以确定权重3的CAC的最大大小，这是一个棘手的问题。在本文中，我们将通过重新讨论模余数环和环多项式环中一个单元的乘性阶的一些性质，给出奇数长度和权重3的紧密/最佳等差CAC的一些显式序列。

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