Modified linear block code with code rate 1/2 and less than 1/2

Abstract

This work proposes a new code which can be considered as the modification of linear block code with code rate 1/2 and less than 1/2. The theme of the work is derived from the controllability of discrete time system and accordingly the generator matrix is chosen. A procedure is proposed in this paper which can transform the lower dimensional space to higher dimensional space. A Code is subspace of a space, so, with the help of addition of an equivalent identity matrix, between I (Identity matrix) and P (Parity Check) matrix of the established generator matrix, the lower dimensional space is transformed to higher dimensional space. Multiple numbers of errors could be corrected with the help of this designed code. It is also shown that decoding of this code can be established using the methods like syndrome detection and nearest neighbor decoding. The proposed code satisfies all the bounds. The minimum distance d, which could be achieved here, is at least equal to the total no. of information bits which enhances its error correction capacity.

Keywords:

• Basis Vectors
• Discrete time system
• Generator Matrix
• Parity Check Matrix
• Syndrome decoding
• Nearest neighbor Decoding
• Bounds
• Minimum distance
• Code word