一个关于Hamming图识别的猜想的解决
Solving a Conjecture on Identification in Hamming Graphs
摘要：
自Karpovsky、Chakrabarty和Levitin在1998年提出图中的识别码以来，图中的识别码得到了广泛的研究。特别地，关于二元超立方体，即Hamming图$K_2^n$，已有很多结果。2008年，Gravier等人开始研究$K_q^2$中的识别。Goddard和Wash在2013年研究了一般Hamming图$K_q^n$中的识别码。例如，他们指出，任何$q$和$n\geq3$的$γ^{ID}(K_q^n)\leq q^{n-1}$。此外，他们还强调了$γ^{ID}(K_q^3)=q^2$。在这篇文章中，我们表明，$γ^{ID}(K_q^3)\leq q^2-q/4$当$q$是一个4的权力，反驳了猜想。我们的方法是基于递归使用合适的设计。Goddard和Wash也给出了下面的下限$γ^{ID}(K_q^3)\ge q^2-q\sqrt{q}$。我们将此界改进为$γ^{ID}(K_q^3)\ge q^2-\frac{3}{2} q$。用于获得识别码的下限的常规方法在这里没有帮助。因此，我们提供了一个不同的技术戈达德和华盛顿的方法的建设。此外，当$q$是素数幂时，我们将上述的界$γ^{ID}(K_q^n)\leq q^{n-1}$改进为$γ^{ID}(K_q^n)\leq q^{n-k}$（对于$n=3\frac{q^k-1}{q-1}$）.对于这个界，我们利用有限域上合适的线性码和一类密切相关的码，即自定位控制码。此外，我们还证明了自定位控制码满足与上述猜想相关的结果$γ^{SLD}(K_q^3)=q^2$.
英文摘要：
Identifying codes in graphs have been widely studied since their introduction by Karpovsky, Chakrabarty and Levitin in 1998. In particular, there are a lot of results regarding the binary hypercubes, that is, the Hamming graphs $K_2^n$. In 2008, Gravier et al. started investigating identification in $K_q^2$. Goddard and Wash, in 2013, studied identifying codes in the general Hamming graphs $K_q^n$. They stated, for instance, that $γ^{ID}(K_q^n)\leq q^{n-1}$ for any $q$ and $n\geq3$. Moreover, they conjectured that $γ^{ID}(K_q^3)=q^2$. In this article, we show that $γ^{ID}(K_q^3)\leq q^2-q/4$ when $q$ is a power of four, disproving the conjecture. Our approach is based on the recursive use of suitable designs. Goddard and Wash also gave the following lower bound $γ^{ID}(K_q^3)\ge q^2-q\sqrt{q}$. We improve this bound to $γ^{ID}(K_q^3)\ge q^2-\frac{3}{2} q$. The conventional methods used for obtaining lower bounds on identifying codes do not help here. Hence, we provide a different technique building on the approach of Goddard and Wash. Moreover, we improve the above mentioned bound $γ^{ID}(K_q^n)\leq q^{n-1}$ to $γ^{ID}(K_q^n)\leq q^{n-k}$ for $n=3\frac{q^k-1}{q-1}$ when $q$ is a prime power. For this bound, we utilize suitable linear codes over finite fields and a class of closely related codes, namely, the self-locating-dominating codes. In addition, we show that the self-locating-dominating codes satisfy the result $γ^{SLD}(K_q^3)=q^2$ related to the above conjecture.
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