New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming
نویسندگان
چکیده
We give a new upper bound on the maximum size Aq(n, d) of a code of word length n and minimum Hamming distance at least d over the alphabet of q ≥ 3 letters. By blockdiagonalizing the Terwilliger algebra of the nonbinary Hamming scheme, the bound can be calculated in time polynomial in n using semidefinite programming. For q = 3, 4, 5 this gives several improved upper bounds for concrete values of n and d. This work is related to [6], where a similar approach is used to derive upper bounds for binary codes.
منابع مشابه
Upper bounds for ternary constant weight codes from semidefinite programming and representation theory
In this thesis we use a semidefinite programming approach to find explicit upper bounds on the size of ternary constant weight codes with prescribed minimum distance d. By constructing a graph Γ = (X,E), on the set, X, of all possible ternary words of weight w letting {x, y} ∈ E ⇔ 0 < dH(x, y) < d, we can view this problem as a special case of the stable set problem. Using symmetry, the constra...
متن کاملSemidefinite programming for permutation codes
We initiate study of the Terwilliger algebra and related semidefinite programming techniques for the conjugacy scheme of the symmetric group Sym(n). In particular, we compute orbits of ordered pairs on Sym(n) acted upon by conjugation and inversion, explore a block di-agonalization of the associated algebra, and obtain improved upper bounds on the size M (n, d) of permutation codes of lengths u...
متن کاملStrengthened semidefinite programming bounds for codes
We give a hierarchy of semidefinite upper bounds for the maximum size A(n,d) of a binary code of word length n and minimum distance at least d. At any fixed stage in the hierarchy, the bound can be computed (to an arbitrary precision) in time polynomial in n; this is based on a result of de Klerk et al. (Math Program, 2006) about the regular ∗-representation for matrix ∗-algebras. The Delsarte ...
متن کاملSemidefinite bounds for nonbinary codes based on quadruples
For nonnegative integers q, n, d , let Aq(n, d) denote themaximum 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 ∑ v∈[q]n x({v}), where x is a function C4→R+ such that ...
متن کاملA Graph-Theoretic Approach to Bounds for Error-Correcting Codes CIMPA-UNESCO-PHILIPPINES Summer School on Semidefinite Programming in Algebraic Combinatorics
In these notes, we address bounds for error-correcting codes. Our approach is from the viewpoint of algebraic graph theory. We therefore begin with a review of the algebraic structure of the Hamming graph, focusing on the binary case. We then derive Delsarte’s linear programming bound and explore some applications of it. In the second part of the notes, we introduce Terwilliger’s subconstituent...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- J. Comb. Theory, Ser. A
دوره 113 شماره
صفحات -
تاریخ انتشار 2006