Hi, I need some help on a problem regarding coset codes of Reed-Solomon codes (using GF(2^m)). I will quickly describe the problem. Define the same_symbol_weight of a codeword c as the maximum number of same symbols contained in c. Define the same_symbol_weight of a codebook C as the number of same symbols of c_s, where c_s is a codeword in C containing the maximum number of same symbols. As an example, the same_symbol_weight of the all-zero codeword of an (n,k) RS code is equal to n. Also, the same_symbol_weight of the all e codeword (e \in GF(2^m)) is equal to n. Thus the same_symbol_weight of an (n,k) RS codebook is equal to n. Through simulation, I have found that the minimum same_symbol_weight of (C + h) is equal to k. I also found that coset leaders that will give a same_symbol_weight of k can be constructed by either (x - \alpha^1)(x - \alpha^2)...(x-\alpha^{n-k-1}) or (x - \alpha^2)(x - \alpha^3)...(x - \alpha^{n-k}), given that the RS generator polynomial is (x - \alpha^1)(x - \alpha^2)...(x - \alpha^{n-k}). [I assume that if the generator polynomial is constructed with the first factor as (x - \alpha^{i+1}), the coset leaders should be adjusted accordingly (leaving out the first or last factor) and having the same results, although I haven't test it yet...] How can I prove that the above coset leaders will always give a same_symbol_weight(C+h) of k? (C is the RS code, h is one of the coset leaders shown, and same_symbol_weight(C+h) is the same symbol weight of the coset code (C+h)). Any help will be greatly appreciated, Jaco Versfeld

# Coset leader of a coset code of RS codes

Started by ●September 21, 2005

Reply by ●September 23, 20052005-09-23

<jaco.versfeld@gmail.com> asked in message news:1127307172.341174.298490@g44g2000cwa.googlegroups.com...> I also found that coset leaders that will give > a same_symbol_weight of k can be constructed by either (x - \alpha^1)(x > - \alpha^2)...(x-\alpha^{n-k-1}) or (x - \alpha^2)(x - \alpha^3)...(x - > \alpha^{n-k}), given that the RS generator polynomial is (x - > \alpha^1)(x - \alpha^2)...(x - \alpha^{n-k}). > > How can I prove that the above coset leaders will always give a > same_symbol_weight(C+h) of k? (C is the RS code, h is one of the coset > leaders shown, and same_symbol_weight(C+h) is the same symbol weight of > the coset code (C+h)).Here is something to start you on your way to the proof. Your h (defined in either of the two ways) is a codeword in (in fact, the generator polynomial of) an (n, k+1) supercode of C. This supercode is also an RS code. Thus, h, a polynomial of degree n-k-1, has weight n-k (i.e. all coefficients are nonzero) and hence 0 + h is a word of same-symbol-weight k (there are k zeroes in it). So the same-symbol-weight of C + h is at least k. Now prove that it cannot be more than k..... --Dilip Sarwate

Reply by ●September 28, 20052005-09-28