Semidefinite programming bounds for Lee codes

For $q,n,d \in \mathbb{N}$, let $A_q^L(n,d)$ denote the maximum cardinality of a code $C \subseteq \mathbb{Z}_q^n$ with minimum Lee distance at least $d$, where $\mathbb{Z}_q$ denotes the cyclic group of order $q$. We consider a semidefinite programming bound based on triples of codewords, which bound can be computed efficiently using symmetry reductions, resulting in several new upper bounds on $A_q^L(n,d)$. The technique also yields an upper bound on the independent set number of the $n$-th strong product power of the circular graph $C_{d,q}$, which number is related to the Shannon capacity of $C_{d,q}$. Here $C_{d,q}$ is the graph with vertex set $\mathbb{Z}_q$, in which two vertices are adjacent if and only if their distance (mod $q$) is strictly less than $d$. The new bound does not seem to improve significantly over the bound obtained from Lov\'asz theta-function, except for very small $n$.Comment: 14 pages. arXiv admin note: text overlap with arXiv:1703.0517

Quasi-Perfect Lee Codes of Radius 2 and Arbitrarily Large Dimension

A construction of two-quasi-perfect Lee codes is given over the space ?np for p prime, p ? ±5 (mod 12), and n = 2[p/4]. It is known that there are infinitely many such primes. Golomb and Welch conjectured that perfect codes for the Lee metric do not exist for dimension n ? 3 and radius r ? 2. This conjecture was proved to be true for large radii as well as for low dimensions. The codes found are very close to be perfect, which exhibits the hardness of the conjecture. A series of computations show that related graphs are Ramanujan, which could provide further connections between coding and graph theories

Semidefinite programming bounds for Lee codes

For q, n, d ∈N, let ALq(n,d) denote the maximum cardinality of a code C ⊆ Znq with minimum Lee distance at least d, where Zq denotes the cyclic group of order q. We consider a semidefinite programming bound based on triples of codewords, which bound can be computed efficiently using symmetry reductions, resulting in several new upper bounds on ALq(n,d). The technique also yields an upper bound on the independent set number of the nth strong product power of the circular graph Cd,q, which number is related to the Shannon capacity of Cd,q. Here Cd,q is the graph with vertex set Zq, in which two vertices are adjacent if and only if their distance (mod q) is strictly less than d. The new bound does not seem to improve significantly over the bound obtained from Lovász theta-function, except for very small n