Semidefinite bounds for nonbinary codes based on quadruples

For nonnegative integers $q,n,d$, let $A_q(n,d)$ 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 $A_q(n,d)$. For any $k$, let $\CC_k$ be the collection of...

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Published in	arXiv.org
Main Authors	Litjens, Bart, Polak, Sven, Schrijver, Alexander
Format	Paper Journal Article
Language	English
Published	Ithaca Cornell University Library, arXiv.org 08.02.2016
Subjects	Codes Integers Mathematics - Combinatorics Mathematics - Optimization and Control Mathematics - Representation Theory Polynomials Semidefinite programming Upper bounds
Online Access	Get full text
ISSN	2331-8422
DOI	10.48550/arxiv.1602.02531

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Abstract	For nonnegative integers $q,n,d$, let $A_q(n,d)$ 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 $A_q(n,d)$. For any $k$, let $\CC_k$ be the collection of codes of cardinality at most $k$. Then $A_q(n,d)$ is at most the maximum value of $\sum_{v\in[q]^n}x(\{v\})$, where $x$ is a function $\CC_4\to R_+$ such that $x(\emptyset)=1$ and $x(C)=0$ if $C$ has minimum distance less than $d$, 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 $n$. It yields the new upper bounds $A_4(6,3)\leq 176$, $A_4(7,4)\leq 155$, $A_5(7,4)\leq 489$, and $A_5(7,5)\leq 87$.
AbstractList	For nonnegative integers $q,n,d$, let $A_q(n,d)$ 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 $A_q(n,d)$. For any $k$, let $\CC_k$ be the collection of codes of cardinality at most $k$. Then $A_q(n,d)$ is at most the maximum value of $\sum_{v\in[q]^n}x(\{v\})$, where $x$ is a function $\CC_4\to R_+$ such that $x(\emptyset)=1$ and $x(C)=0$ if $C$ has minimum distance less than $d$, 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 $n$. It yields the new upper bounds $A_4(6,3)\leq 176$, $A_4(7,4)\leq 155$, $A_5(7,4)\leq 489$, and $A_5(7,5)\leq 87$. Designs, Codes and Cryptography, 84 (1) (2017), 87-100 For nonnegative integers $q,n,d$, let $A_q(n,d)$ 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 $A_q(n,d)$. For any $k$, let $\CC_k$ be the collection of codes of cardinality at most $k$. Then $A_q(n,d)$ is at most the maximum value of $\sum_{v\in[q]^n}x(\{v\})$, where $x$ is a function $\CC_4\to R_+$ such that $x(\emptyset)=1$ and $x(C)=0$ if $C$ has minimum distance less than $d$, 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 $n$. It yields the new upper bounds $A_4(6,3)\leq 176$, $A_4(7,4)\leq 155$, $A_5(7,4)\leq 489$, and $A_5(7,5)\leq 87$.
Author	Polak, Sven Litjens, Bart Schrijver, Alexander
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BackLink	https://doi.org/10.1007/s10623-016-0216-5$$DView published paper (Access to full text may be restricted) https://doi.org/10.48550/arXiv.1602.02531$$DView paper in arXiv
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Title	Semidefinite bounds for nonbinary codes based on quadruples
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Abstract	For nonnegative integers \(q,n,d\), let \(A_q(n,d)\) 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 \(A_q(n,d)\). For any \(k\), let \(\CC_k\) be the collection of codes of cardinality at most \(k\). Then \(A_q(n,d)\) is at most the maximum value of \(\sum_{v\in[q]^n}x(\{v\})\), where \(x\) is a function \(\CC_4\to R_+\) such that \(x(\emptyset)=1\) and \(x(C)=0\) if \(C\) has minimum distance less than \(d\), 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 \(n\). It yields the new upper bounds \(A_4(6,3)\leq 176\), \(A_4(7,4)\leq 155\), \(A_5(7,4)\leq 489\), and \(A_5(7,5)\leq 87\).
AbstractList	For nonnegative integers \(q,n,d\), let \(A_q(n,d)\) 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 \(A_q(n,d)\). For any \(k\), let \(\CC_k\) be the collection of codes of cardinality at most \(k\). Then \(A_q(n,d)\) is at most the maximum value of \(\sum_{v\in[q]^n}x(\{v\})\), where \(x\) is a function \(\CC_4\to R_+\) such that \(x(\emptyset)=1\) and \(x(C)=0\) if \(C\) has minimum distance less than \(d\), 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 \(n\). It yields the new upper bounds \(A_4(6,3)\leq 176\), \(A_4(7,4)\leq 155\), \(A_5(7,4)\leq 489\), and \(A_5(7,5)\leq 87\). Designs, Codes and Cryptography, 84 (1) (2017), 87-100 For nonnegative integers $q,n,d$, let $A_q(n,d)$ 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 $A_q(n,d)$. For any $k$, let $\CC_k$ be the collection of codes of cardinality at most $k$. Then $A_q(n,d)$ is at most the maximum value of $\sum_{v\in[q]^n}x(\{v\})$, where $x$ is a function $\CC_4\to R_+$ such that $x(\emptyset)=1$ and $x(C)=0$ if $C$ has minimum distance less than $d$, 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 $n$. It yields the new upper bounds $A_4(6,3)\leq 176$, $A_4(7,4)\leq 155$, $A_5(7,4)\leq 489$, and $A_5(7,5)\leq 87$.
Author	Polak, Sven Litjens, Bart Schrijver, Alexander
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