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On the linear programming bound for linear Lee codes.

Helena Astola1, Ioan Tabus1

  • 1Department of Signal Processing, Tampere University of Technology, 33101 Tampere, Finland.

Springerplus
|March 31, 2016
PubMed
Summary

Researchers introduce new equality constraints for linear Lee codes by leveraging an invariance property. This enhances linear programming, enabling faster computation of bounds for large linear codes.

Keywords:
Lee codesLee-compositionsLee-numbersLinear codesLinear programming bound

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Area of Science:

  • Coding Theory
  • Algebraic Combinatorics
  • Computational Mathematics

Background:

  • Linear Lee codes are fundamental in coding theory, with their properties often analyzed using linear programming.
  • The Lee association scheme and its eigenvalues (Lee-numbers) play a crucial role in understanding these codes.
  • Existing linear programming formulations can be computationally intensive for large parameter values.

Purpose of the Study:

  • To introduce novel equality constraints for the linear programming problem of linear Lee codes.
  • To leverage an invariance property of Lee-compositions and the action of a multiplicative group.
  • To develop a more computationally efficient method for determining bounds of linear Lee codes.

Main Methods:

  • Formulating an invariance property of Lee-compositions in terms of group action.
  • Analyzing properties of sums of Lee-numbers (eigenvalues of the Lee association scheme).
  • Developing a compact linear programming formulation using additional equality constraints.

Main Results:

  • Demonstrated an invariance-type property of Lee-compositions.
  • Established useful properties of specific sums of Lee-numbers.
  • Formulated a compact linear programming problem for linear Lee codes.
  • Achieved faster execution for computing bounds of linear codes with large parameters.

Conclusions:

  • The proposed method significantly enhances the efficiency of computing bounds for linear Lee codes.
  • The new formulation allows for the rapid analysis of linear codes with large parameter values.
  • This research offers a more computationally tractable approach to linear Lee code optimization.