Abstract Algebra: Theory and Applications · Problem 8.7.1
Isomorphisms — Thomas Judson
Mathematics — Abstract Algebra · linear-code-implementation
Problem
Write a program to implement a -linear code. Your program should be able to encode and decode messages using coset decoding. Once your program is written, write a program to simulate a binary symmetric channel with transmission noise. Compare the results of your simulation with the theoretically predicted error probability.Intuitive explanation
A -linear code lets you add 4 redundant bits to every 12 message bits, creating a structured way to detect and correct errors. The key insight is that coset decoding works by partitioning all possible received words into cosets of the code—you decode by finding which codeword is closest to what you received. Simulating a noisy channel and comparing real error rates to theory tests whether your code's error-correction capability actually matches the math.
Intuitive explanation
A -linear code lets you add 4 redundant bits to every 12 message bits, creating a structured way to detect and correct errors. The key insight is that coset decoding works by partitioning all possible received words into cosets of the code—you decode by finding which codeword is closest to what you received. Simulating a noisy channel and comparing real error rates to theory tests whether your code's error-correction capability actually matches the math.