Additive perfect codes in Doob graphs

Abstract

The Doob graph D(m, n) is the Cartesian product of m > 0 copies of the Shrikhande graph and n copies of the complete graph of order 4. Naturally, D(m, n) can be represented as a Cayley graph on the additive groupA set of vertices of D(m, n) is called an additive code if it forms a subgroup of this group. We construct a 3-parameter class of additive perfect codes in Doob graphs and show that the known necessary conditions of the existence of additive 1-perfect codes in D(m, n ′ + n ′′ ) are sufficient. Additionally, two quasi-cyclic additive 1-perfect codes are constructed in D(155, 0 + 31) and D(2667, 0 + 127).

On Dual Codes in the Doob Schemes

“…The motivation for this extension lies in the extended possibility to construct codes with different parameters in the same metric space (as we will see in the next subsection, the coordinates from the last two groups are metrically equivalent in the considered scheme). Examples of exploiting this possibility can be found in the construction of additive 1-perfect codes [9].…”

“…2. It [5], [6], and [9], classes of linear and additive 1-perfect codes in Doob graphs were constructed. By Theorem 2, the code generated by a check matrix of a linear 1-perfect code in D(m, n) has the parameters of the code dual to the linear 1-perfect quaternary Hamming code of length 2m + n (the dual code is a simplex code, or a tight 2-design, see e.g.…”

The Existence of Perfect Codes in Doob Graphs

“…Namely, we show the existence of 1-perfect codes in the Doob graph D(m, n) for all m and n that satisfy the obvious necessary condition: the size 6m + 3n + 1 of a ball of radius 1 divides the number 4 2m+n of vertices. In the previous papers [9], [10], [15], the problem was solved only for the cases when the parameters satisfy additional conditions admitting the existence of linear or additive perfect codes, or for small values of m.…”

“…Namely, we show the existence of 1-perfect codes in the Doob graph D(m, n) for all m and n that satisfy the obvious necessary condition: the size 6m + 3n + 1 of a ball of radius 1 divides the number 4 2m+n of vertices. In the previous papers [9], [10], [15], the problem was solved only for the cases when the parameters satisfy additional conditions admitting the existence of linear or additive perfect codes, or for small values of m.The class of Doob graphs is a class of distance-regular graphs of unbounded diameter, and the problem considered can be viewed in the general context of the problem of existence of perfect codes in distanceregular graphs. We mention some known results in this area, mainly concentrating on the distance-regular graphs important for coding theory.…”

“…In [10], infinite series of perfect codes in Doob graphs were obtained. In particular, it was shown that the necessary condition 2m + n = (4 k − 1)/3 is sufficient if m < n − o(2m + n); the class of linear perfect codes was completely characterized; a class of additive perfect codes was constructed and necessary conditions on m, n ′ , n ′′ for the existence of additive perfect codes in D(m, n ′ + n ′′ ) were obtained (in a recent work [15], it was shown that those conditions are also sufficient).…”