Generalized Hamming Weights for Linear Codes Iskra N´u˜nez Estefan Ortiz University of Texas at El Paso St. Mary’s University Alicia Urdapilleta Mills College July 27, 2001 Abstract Error control codes are widely used to increase the reliability of transmission of information over various forms of communications channels. The Hamming weight of a codeword is the number of nonzero entries in the word; the weights of the words in a linear code determine the error-correcting capacity of the code. th The r generalized Hamming weight for a linear code C, denoted by dr(C), is the minimum of the support sizes for r-dimensional subcodes of C. For instance, d1(C) equals the traditional minimum Hamming weight of C. In 1991, Feng, Tzeng, and Wei proved that the second generalized Hamming weight d2(C) = 8 m for all double-error correcting BCH(2 , 5) codes. We study d3(C) and higher Hamming weights for BCH(2m, 5) codes by a close examination of the words of weight 5. 1 Introduction The generalized Hamming weights of a linear code were introduced by Wei in 1991. The rth generalized Hamming weight is, by definition, the minimum support size of r- dimensional subcodes of C. Soon after, Feng, Tzeng, and Wei studied the generalized Hamming weights for BCH and other cyclic codes. They showed in particular that the second generalized Hamming weight of the double-correcting binary BCH(2m, 5) codes is d2 = 8. They accomplished this by studying the subspaces generated by pairs of words of weight five in the code. Motivated by these findings, in this paper we continue the study of the dr(C) for C = BCH(2m, 5), by considering the cases r ≥ 3. First, we review several basic properties of the generalized weights (Theorems 1,2,3 below) which help us to understand dr(C) for linear codes. Using these theorems, we develop algorithms that allow us to compute the complete weight hierarchies of several classes of codes such as the binary Hamming and Golay codes. We then turn our attention to the BCH(2m, 5) codes and prove a result (Theorem 6) that counts the words of weight 5 with ones in two specified locations in the case of odd m. By considering the possible overlaps of 35 the positions of the 1’s in sets of three words of weight 5, we deduce that d3(C) = 10 for the BCH(2m, 5) codes with r ≥ 4. Finally, we look into the higher generalized Hamming weighs of the BCH(2m, 5) codes and propose questions for further research. 2 Basic Properties Our focus is linear binary codes. We begin by stating the some fundamental properties and definitions. Definition 1 Let C be an [n, k] code with D ⊂ C a subcode. The support of D, denoted χ(D), is the set of not-always-zero bit positions of D, i.e., χ(D) := {i : ∃(x1, x2, ··· , xn) ∈ D, xi 6= 0} Using this definition, an [n, k] code is a binary linear code of rank k and support size ≤ n. The rank is the number of non-zero linearly independent rows of a matrix. Example 1. Let C = h10011, 01010, 00101i. The matrix 1 0 0 1 1 G = 0 1 0 1 0 0 0 1 0 1 whose rows are the generators for C is already in row-reduced echelon form, so these vectors are a basis for C. We see that χ(C) = {1, 2, 3, 4, 5}, since there are 1’s in all of those positions in some word in C. Let D be a one-dimensional subcode of C. The size of the support of D equals the Hamming weight of the non-zero codeword in D. Hence, the minimum distance of C is the minimum of |χ(D)| as D ranges over all one-dimensional subcodes of C. Wei generalized this definition. th Definition 2 The r generalized Hamming weight of C, denoted dr(C), is the smallest support of an r-dimensional subcode of C. That is, dr(C) := min{|χ(D)| : D is a subcode of C with rank r} To better understand this definition we take the already defined linear code C from the previous example and its generator matrix G. Example 2. Referring to Example 1, we see that d1(C) = 2 equals the usual min- imum Hamming weight of C. We obtained d2(C) by taking the minimum support size of all two-dimensional subspaces D ⊂ C. The smallest weight is attained for the subspace spanned by the first two rows of G, and d2(C) = minD{|χ(D)|} = 4. There is some codeword in C with a 1 in each of the 5 positions, so d3(C) = 5. We can also assemble all generalized Hamming weights of a code to form the weight hierarchy. 36 Definition 3 The weight hierarchy of a linear code C is defined to be the set of integers {dr(C) : 1 ≤ r ≤ k}. 3 Wei’s Theorems In this section, we collect several general results from [1] about the generalized Ham- ming weights. Wei’s first theorem gives a generalized Singleton bound for dr(C). Theorem 1 ([1], Theorem 1 – Monotonicity) For an [n, k] code C with k > 0, we have 1 ≤ d1(C) < d2(C) < ··· < dk(C) ≤ n. Corollary 1 (Generalized Singleton Bound) For an [n, k] code C, dr(C) ≤ n−k+r. (When r=1, this is the Singleton bound.) Proof: That dr−1(C) ≤ dr(C) follows directly from the definition; it remains to prove that strict inequalities hold. Let D be a subcode with rank(D) = r and |χ(D)| = dr(C). Let i ∈ χ(D) and Di := {x ∈ D : xi = 0}. Then rank (Di) = r − 1 and dr−1(C) ≤ |χ(Di)| ≤ |χ(D)| − 1 = dr(C) − 1. 4 We will write a parity check matrix of an [n, k] linear code as an (n−k)×n matrix. The codewords are then the column vectors x of length n such that Hx = 0. Theorem 2 gives us an alternative way to compute the generalized Hamming weight. First, we define hHi : i ∈ Ii to be the space generated by the column vectors Hi, 1 ≤ i ≤ n. Theorem 2 ([1], Theorem 2) For all r ≤ k, dr(C) = min{|I| : |I| − rank(hHi : i ∈ Ii) ≥ r} Proof: For any I ⊂ {1, 2, ··· , n}, let S(I) = hHi : i ∈ Ii be the space spanned by the columns of H numbered by elements of I. Let ⊥ X S (I) := {x : xi = 0 for all i ∈ I and xiHi = 0}. i∈l Then rank(S(I)) + rank(S⊥(I)) = |I|. Let d equal the quantity on the right-hand side of the equality in the statement of Theorem 2. Let I ⊂ {1, 2, ··· , n} be such that |I| − rank(S(I)) = r and |I| = d. ⊥ ⊥ ⊥ Then rank(S (I)) = r, S (I) is a subcode of C, and dr(C) ≤ |χ(S (I))| ≤ |I| = d. So dr(C) ≤ d. It remains to establish the inequality in the other direction. Let D be a subcode of C with rank(D) = r and |χ(D)| = dr(C). Let I = |χ(D)|, then D ⊂ S⊥(I). But rank (S(I)) = |I| − rank(S⊥(I)) ≤ |I| − r, so |I| − rank(S(I)) ≥ r. 0 ⊥ ⊥ Suppose |I| − rank(S(I)) = r > r. Then D 6= S (I), and dr(C) ≤ |χ(S (I)| ≥ |I|, a contradiction. Hence |I| − rank(S(I)) = r, and d ≤ dr(C). 4 37 Example 3 We consider the generator matrix G from the previous examples, and construct a parity check matrix H by the usual algorithm. 1 0 0 1 1 G = 0 1 0 1 0 0 0 1 0 1 so 1 1 0 1 0 H = . 1 0 1 0 1 Let I be any subset of {1, 2 ··· , n}. Consider the vector subspace generated by the columns Hi for i ∈ I. We seek the I that satisfy the given condition |I| − rank(hHi : i ∈ Ii) ≥ r and for which |I| is minimized. For instance with r = 1, we see that the smallest sets of columns satisfying |I| − rank(hHi : i ∈ Ii) ≥ 1 are I = {2, 4} or I = {3, 5}. Hence d1(C) = 2. Similarly, the smallest sets of columns for which |I| − rank(hHi : i ∈ Ii) ≥ 2 have size |I| = 4 (I = {1, 2, 3, 4} is one such set). So d2(C) = 4. Finally, d3(C) = 5. Theorem 3 gives a relation between the generalized weights of C and the general- ized weights of the dual code C⊥. Theorem 3 ([1], Theorem 3 – Duality) Let C be an [n, k] code, then ⊥ {dr(C) : 1 ≤ r ≤ k} = {1, 2, ··· , n}\{n + 1 − dr(C ) : 1 ≤ r ≤ n − k} ⊥ Proof: It suffices to prove that for any r with 1 ≤ r < n−k, n+1−dr(C ) ∈/ {di(C): ⊥ ⊥ 1 ≤ i ≤ k}. First we prove that dr(C) ≤ n − dr(C ), where t = k + r − dr(C ). ⊥ ⊥ Let D be a subcode of C with rank(D) = r and |χ(D)| = dr(C ). There exists a parity-check matrix H for C where first r row are vectors in D and the last n − k − r T rows are not. The column vectors {Hi : i∈ / χ(D) have their first r coordinates T T zero. Hence rank (hHi : i∈ / χ(D)i) =column rank (hHi : i∈ / χ(D)i) ≤ row th rank (hRi : r + 1 ≤ i ≤ n − ki) = n − k − r. (Ri is the i row-vector of H.) ⊥ By Theorem 2, letting I = {1, 2, ··· , n}\χ(D), we obtain, I = n − dr(C ), with ⊥ t = |I| − (n − k − r) = k + r − dr(C ).