On Matrix Representation of Extension Field GF(p) and Its Application in Vector Linear Network Coding

Area of Science:

• Coding Theory
• Finite Field Arithmetic
• Network Coding

Background:

• Standard matrix representations of finite fields GF(pL) over GF(p) enable arithmetic operations.
• Existing implementations of coding schemes over GF(2L) use matrix representations but overlook correlations between matrices representing different field elements.

Purpose of the Study:

• Generalize classical matrix representation results from GF(2L) to GF(pL).
• Reveal and leverage inherent correlations among matrices representing different powers of GF(pL) elements.
• Improve the efficiency of vector linear network coding (LNC) schemes.

Main Methods:

• Generalization of matrix representation for GF(pL) over GF(p).
• Analysis of inherent correlations among matrices representing powers of field elements.
• Development of a pre-stored lookup table based on matrix correlations.

Main Results:

• A generalized matrix representation for GF(pL) over GF(p) is established.
• Inherent correlations among matrices representing different powers of GF(pL) elements are identified and clarified.
• A more compact lookup table for matrix representation is devised, reducing storage requirements.
• Theoretical results demonstrating the advantages of binary matrix representation in vector LNC are derived.

Conclusions:

• The generalized matrix representation and identified correlations offer a more transparent understanding of matrix-based finite field arithmetic.
• The proposed lookup table enhances the efficiency of vector LNC implementations.
• The findings underscore the benefits of binary matrix representations in advanced coding schemes.