25 research outputs found
Semidefinite bounds for nonbinary codes based on quadruples
For nonnegative integers , let denote the maximum
cardinality of a code of length over an alphabet with letters and
with minimum distance at least . We consider the following upper bound on
. For any , let \CC_k be the collection of codes of cardinality
at most . Then is at most the maximum value of
, where is a function \CC_4\to R_+ such that
and if has minimum distance less than , and
such that the \CC_2\times\CC_2 matrix (x(C\cup C'))_{C,C'\in\CC_2} is
positive semidefinite. By the symmetry of the problem, we can apply
representation theory to reduce the problem to a semidefinite programming
problem with order bounded by a polynomial in . It yields the new upper
bounds , , , and
Semidefinite bounds for nonbinary codes based on quadruples
For nonnegative integers q, n, d, let Aq(n, d) denote the maximum cardinality of a code
of length n over an alphabet [q] with q letters and with minimum distance at least d. We consider
the following upper bound on Aq(n, d). For any k, let Ck be the collection of codes of cardinality
at most k. Then Aq(n, d) is at most the maximum value of Pvβ[q]n x({v}), where x is a function
C4 β R+ such that x(β
) = 1 and x(C) = 0 if C has minimum distance less than d, and such that
the C2 ΓC2 matrix (x(C βͺCβ²))C,Cβ²βC2 is positive semidefinite. By the symmetry of the problem, we
can apply representation theory to reduce the problem to a semidefinite programming problem with
order bounded by a polynomial in n. It yields the new upper bounds A4(6, 3) β€ 176, A4(7, 4) β€ 155,
A5(7, 4) β€ 489, and A5(7, 5) β€ 87
Semidefinite bounds for mixed binary/ternary codes
For nonnegative integers and , let denote the
maximum cardinality of a code of length , with binary
coordinates and ternary coordinates (in this order) and with minimum
distance at least . For a nonnegative integer , let
denote the collection of codes of cardinality at most . For , define . Then is upper bounded by the maximum
value of , where is a function
such that and if has minimum distance less than , and such that the matrix is positive semidefinite for each
. By exploiting symmetry, the semidefinite programming
problem for the case is reduced using representation theory. It yields
new upper bounds that are provided in tablesComment: 12 pages; some typos have been fixed. Accepted for publication in
Discrete Mathematic
Semidefinite programming bounds for Lee codes
For , let denote the maximum cardinality
of a code with minimum Lee distance at least ,
where denotes the cyclic group of order . 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 . The technique also yields an upper bound on the
independent set number of the -th strong product power of the circular graph
, which number is related to the Shannon capacity of . Here
is the graph with vertex set , in which two vertices
are adjacent if and only if their distance (mod ) is strictly less than .
The new bound does not seem to improve significantly over the bound obtained
from Lov\'asz theta-function, except for very small .Comment: 14 pages. arXiv admin note: text overlap with arXiv:1703.0517