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Uniqueness of codes using semidefinite programming.

Andries E Brouwer1, Sven C Polak2

  • 1Amsterdam, The Netherlands.

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Summary
This summary is machine-generated.

This study proves the uniqueness of binary codes achieving specific size bounds for constant weight codes. It also reveals multiple nonisomorphic codes meeting the maximum size bound for binary codes with a given minimum distance.

Keywords:
Binary codeCodeGolaySemidefinite programmingUniqueness

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Area of Science:

  • Coding Theory
  • Discrete Mathematics
  • Computer Science

Background:

  • Binary codes are fundamental in information theory and computer science.
  • Determining the maximum size of binary codes with specific parameters (length, distance, weight) is a key challenge.

Purpose of the Study:

  • To establish the uniqueness of codes that achieve previously established upper bounds for constant weight binary codes.
  • To investigate the structure and classification of codes that achieve the maximum size bound for binary codes with a given minimum distance.

Main Methods:

  • Leverages semidefinite programming techniques to analyze code properties.
  • Employs combinatorial methods for the classification of specific code families.

Main Results:

  • Demonstrates the uniqueness of codes achieving the bounds A(n, d, w) under certain conditions.
  • Identifies multiple nonisomorphic codes that attain the bound A(n, d).
  • Provides a classification of these maximal codes when the minimum distance is a multiple of 4.

Conclusions:

  • The findings contribute to a deeper understanding of the structure and enumeration of optimal binary codes.
  • Uniqueness results simplify the study of extremal codes, while classification provides concrete examples.