Optimal equi-difference conflict-avoiding codes of odd length and weight three
Optimal equi-difference conflict-avoiding codes of odd length and weight three
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DOI：
10.1016/j.ffa.2013.11.001
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发表时间：
2014-03
期刊：
影响因子：
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通讯作者：
Yiling Lin;Miwako Mishima;Junya Satoh;Masakazu Jimbo
中科院分区：
文献类型：
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作者：
Yiling Lin;Miwako Mishima;Junya Satoh;Masakazu Jimbo
冲突避免码（CAC）被称为用于在没有反馈的情况下在冲突信道上传输数据分组的协议序列。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，并且在过去的几年中，已经由几位研究人员解决了偶数长度和重量3以及构造。对于奇数码长，在Momihara（2007）中可以找到权重为3的“紧等差”CAC存在的充分必要条件，但该条件相当复杂，因此只有少数明确的码长序列是已知的。最近，Fu et al.（2013）以不同的方式重述了Momihara（2007）给出的条件，该条件要求对m的每个素因子p检查2模p的乘法子阶。同时，Ma et al.（2013）提出了奇素数长度为p且权重为3的最优等差CAC和最优紧CAC的构造，并公式化了此类最优码的大小。然而，为了使它们的公式具有实际意义，−（2）p <$（2）p的陪集的数目仍然需要确定，其中（2）p是Z p <$的乘法子群，生成元为2。此外，他们的最佳紧CAC的建设强加了一定的条件。这意味着，即使将我们自己限制到奇素数长度，以提供一系列奇码长度，对于这些奇码长度，可以确定权重为3的CAC的最大大小，这也是一个苛刻的问题。本文通过重新考察模m剩余环和分圆多项式中单位元乘法阶的一些性质，给出了奇长、奇重3的紧/最优等差CAC的显式序列.
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, ie, 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 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.