{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T18:01:39Z","timestamp":1775066499153,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>In 2013, Goddard and Wash studied identifying codes in the Hamming graphs $K_q^n$. They stated, for instance, that $\\gamma^{ID}(K_q^n)\\leqslant q^{n-1}$ for any $q$ and $n\\geqslant 3$. Moreover, they conjectured that $\\gamma^{ID}(K_q^3)=q^2$. In this article, we show that $\\gamma^{ID}(K_q^3)\\leqslant q^2-q\/4$ when $q$ is a power of four, which disproves the conjecture. Goddard and Wash also gave the lower bound $\\gamma^{ID}(K_q^3)\\geqslant q^2-q\\sqrt{q}$. We improve this bound to $\\gamma^{ID}(K_q^3)\\geqslant q^2-\\frac{3}{2} q$. Moreover, we improve the above mentioned bound $\\gamma^{ID}(K_q^n)\\leqslant q^{n-1}$ to $\\gamma^{ID}(K_q^n)\\leqslant q^{n-k}$ for $n=3\\frac{q^k-1}{q-1}$ and to $\\gamma^{ID}(K_q^n)\\leqslant 3q^{n-k}$ for $n=\\frac{q^k-1}{q-1}$, when $q$ is a prime power. For these bounds, we utilize two classes of closely related codes, namely, the self-identifying and the self-locating-dominating codes. In addition, we show that the self-locating-dominating codes satisfy the result $\\gamma^{SLD}(K_q^3)=q^2$ related to the above conjecture.<\/jats:p>","DOI":"10.37236\/7828","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T07:13:37Z","timestamp":1578640417000},"source":"Crossref","is-referenced-by-count":2,"title":["On a Conjecture Regarding Identification in Hamming Graphs"],"prefix":"10.37236","volume":"26","author":[{"given":"Ville","family":"Junnila","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2688-2166","authenticated-orcid":false,"given":"Tero","family":"Laihonen","sequence":"additional","affiliation":[]},{"given":"Tuomo","family":"Lehtil\u00e4","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2019,6,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i2p45\/7859","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i2p45\/7859","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:12:18Z","timestamp":1579234338000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i2p45"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,21]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,4,5]]}},"URL":"https:\/\/doi.org\/10.37236\/7828","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,6,21]]},"article-number":"P2.45"}}