Groebner basis techniques to compute weight distributions of shortened cyclic codes

Transcription

1 Groebner basis techniques to compute weight distributions of shortened cyclic codes 2nd February 2007 Massimiliano Sala Boole Centre for Research in Informatics, UCC Cork, Ireland. Abstract. Using Gröbner basis techniques, we can exhibit a method to get the distance and weight distribution of cyclic codes and shortened cyclic codes, improving earlier similar results for the distance of cyclic codes. Keywords: cyclic codes, distance, Gröbner bases, weight distribution. 1 Introduction Extensive work has been done to obtain the distance of cyclic codes and their weight distribution (see for example [1, 2, 5, 8, 10, 13, 14, 15, 19, 24]). The use of shortened cyclic codes in many applications has also pushed a parallel study on their properties (see for example [7, 9, 11, 12]). The author has proposed in [22] a method to get the distance of cyclic codes, via the computation of some Gröbner bases. We now show how this method can be improved (by presenting a system where useless solutions have been discarded a priori) and extended to get the weight distribution of cyclic codes. Moreover, a variation of this method allows the computation of the distance and the weight distribution also for shortened cyclic codes. The remainder of this paper is divided into three sections, as follows. Section 2, where we provide our notation. Section 3, where we provide our methods to compute distance and weight distribution for cyclic codes and their shortened codes. Section 4, where we draw some conclusions. 1

2 2 Notation and preliminaries For any a, b Z\{0}, (a, b) denotes their (positive) greatest common divisor. Let K be any field. We denote by χ(k) its characteristic and by K its algebraic closure. Let K[X 1,..., X N ] be any polynomial ring over K and I be any of its ideals. We denote by V(I) its variety, i.e. the set of its solutions V(I) = {ɛ (K) N p(ɛ) = 0, p I}. For any S (K) N, we denote by I K (S) the vanishing ideal on S, i.e. I K (S) = {p K[X 1,..., X N ] p(s) = 0, s S}. For any q we denote by F q the finite field with q elements (so that q is a power of prime χ(f q )). Let n 3 be any integer such that n and q are relatively prime. From now on, q, F q and n are understood. Let α be a primitive n-th root of unity in the splitting field F Q of x n 1 over F q, that is, n 1 (x n 1) = (x α i ). i=0 From now on, α, Q and F Q are understood. Let C be an F q [n, k, d] linear code, for some k 1 and d. Traditionally, integer set {A 0 (C), A 1 (C),..., A n (C)} is called the weight distribution of C, where A i = A i (C) is the number of codewords in C with (Hamming) weight i (0 i n). We recall that A 0 (C) = 1 for any C. Let C be a cyclic [n, k, d] code over F q. The generator polynomial of C will be denoted by g. A cyclic code with defining set S C = {h 1,..., h r } has a parity-check matrix H, which is the standard parity-check matrix, as follows 1 α h 1 α 2h 1 α (n 1)h 1 1 α h 2 α 2h 2 α (n 1)h 2 H = α hr α 2hr α (n 1)hr Observe that the entries of H lie in F Q. Definition 2.1. Let C be an F q cyclic code of length n. Let D be a subset of {0, 1,..., n 1}. Let l = D n 1. We denote by C(D) the code obtained from C by shortening in all positions in D. Clearly, the length of C(D) is n l and C(D) = C if and only if D =, which happens if and only if l = 0. Note that in the previous definition the first position is indicated by position 0 and the last position is indicated by position n 1. This is convenient, as it will be clear later on. 2

3 3 Computing the weight distribution In this section we present our method to compute the weight distribution (and hence the distance) of cyclic codes and their shortened codes. After a few definitions, we state our main result, Theorem 3.3 (without proof), and discuss it, providing some examples and applications. In Subsection 3.1 we give a proof of Theorem 3.3. Definition 3.1. For any i, j such that 1 i j n, we denote by p i,j the following polynomial in F q [z i, z j ]: n 1 p i,j (z i, z j ) = zi h zj n 1 h h=0 = zn i zn j z i z j Definition 3.2. Let C(D) be a shortened cyclic code with the notation adopted in Definition 2.1. Let S C = {h 1,..., h r }. Let w be an integer s.t. 1 w n l. We adopt the following notation: R w = F q [z 1,..., z w, y 1,..., y w ], R w = F Q [z 1,..., z w, y 1,..., y w ], P(D), for the set of w polynomials p i (z i ) = z n i 1 j D (z i α j ), 1 i w Ĵ C(D) (w), for the ideal in R w generated by P (D) and the following three sets of polynomials: {y q 1 1 1,..., y q 1 w 1} {p 1,2 (z 1, z 2 ),..., p i,j (z i, z j ),..., p w 1,w (z w 1, z w )} {y 1 z h y wz h 1 w,..., y 1 z hr y wz hr w } η(d)[w], for the number of solutions of ĴC(D)(w), i.e. for V(ĴC(D)(w)). We are ready for our main theorem. Theorem 3.3. Let C be an F q [n, k, d ] cyclic code. Let D be a subset of {0, 1,..., n 1} with 0 D = l n 1. Let C(D) be the corresponding shortened code. Let 1 w n l. Then A w (C(D)) = η(d)[w] w! When there is no shortening, we have D = and P (D) becomes P ( ) = {z n 1 1,..., zn w 1}, according to the standard convention that the product of a zero number of factors outputs 1 (Π = 1). In this case we denote by Ĵ C (w) the ideal Ĵ C( ) (w). Similarly, η[w] = η( )[w]. It is important to note that ideal ĴC(w) lies in the smaller ring R w and hence computations will be significantly easier. We thus get the following corollary for cyclic codes.. 3

4 Corollary 3.4. Let C be an F q [n, k, d] cyclic code. Let 1 w n. Then A w (C) = η[w] w! Since ĴC(D)(w) is clearly radical and the number of solutions of an ideal I are directly computed from any Groebner basis of I (see [3] for an efficient technique), we can easily describe an algorithm to compute the weight distribution (and hence the distance) of a (shortened) cyclic code.. Algorithm 3.5. Input An F q shortened cyclic code C(D) of length n l. An integer w s.t. 1 w n l. Output The element A w of the weight distribution of C(D). Step 1 Construct the associated system ĴC(D)(w). Step 2 Compute the Gröbner basis Ĝ(w) of the associated ideal. Step 3 Use Ĝ(w) to get the number η(d)[w] of solutions of ĴC(w). Finally output η(d)[w] w! In the binary case conditions {y q 1 i = 1} become trivial, since we get {y i = 1}, and so we can discard variables {y i }, rewriting ideal Ĵ C (D)(w) in polynomial ring T w = Z 2 [z 1,..., z w ] or T w = F Q [z 1,..., z w ], as generated by Ĵ C(D) (w) =< {z h zh 1 w,..., z hr zhr w } P (D) {p i,j (z i, z j )}, where as usual S C = {h 1,..., h r }. We now provide applications of this algorithm, with some remarks and improvements in some cases, as well as comparisons with previous results. For ease of presentation, we will only give examples in the binary case. We will always use degrevlex as the default ordering on polynomial rings, since it is well-known to be the most efficient. We start with a direct application of Theorem 3.3. Example 3.6. Let C be the binary cyclic code of length 15 which is the dual code of BCH(15, 5). We can take S C = {0, 5, 7} as a defining set for C. Suppose that D = {1, 2, 4}, so that the positions in D correspond to the roots of primitive polynomial x 4 + x + 1. With MAGMA ([16]) it is easy to compute the distance of C(D), which is 4, and the number A 4 (C(D)) of 4

5 its minimum weight words, A 4 (C(D)) = 5. We now want to check A 4, by constructing ideal J = ĴC(D)(4) and applying Algorithm 3.5. Polynomial set P (D) is formed by polynomials of kind p(z) = z15 1 z 4 + z + 1 = z11 + z 8 + z 7 + z 5 + z 3 + z 2 + z + 1, so that J is generated by the following polynomials z 1 5 +z 2 5 +z 3 5 +z 4 5, z 1 7 +z 2 7 +z 3 7 +z 4 7, z z 1 8 +z 1 7 +z 1 5 +z 1 3 +z 1 2 +z 1 +1, z z 2 8 +z 2 7 +z 2 5 +z 2 3 +z 2 2 +z 2 +1, z z 3 8 +z 3 7 +z 3 5 +z 3 3 +z 3 2 +z 3 +1, z z z z z z z 4 + 1, p 1,2 (z 1, z 2 ), p 1,3 (z 1, z 3 ), p 1,4 (z 1, z 4 ), p 2,3 (z 2, z 3 ), p 2,4 (z 2, z 4 ), p 3,4 (z 3, z 4 ). The leading terms of the Gröbner basis Ĝ(4) of J are: {z 1 z 2, z 1 z 3 2, z 2 2 z 3 2, z 1 4, z 1 2 z 4 3, z 2 5, z 2 3 z 4 3, z 3 6, z 4 7, z 3 4 z 4 3 } so that the number of solutions is η(d)[4] = 120 and A 4 is 120/4! = 5. Ideal Ĵ C(D) (4) in Example 3.6 has an interesting property: it lies in T w even if the code is a shortened cyclic code. This is due to the fact that we have shortened in positions exactly corresponding to roots of a polynomial with coefficients in Z 2, so that j D (z i α j ) Z 2 [z i ] and hence p i (z i ) Z 2 [z i ]. It is well-known that this happens if and only if D is a union of cyclotomic sets. If D contains only a portion of a conjugacy class, then we are forced to use equations P (D) with coefficients in F Q rather than in Z 2. This is unfortunate, because handling coefficients in a bigger field is more computationally costly and is equivalent to adding another variable to the Gröbner basis computation. However, there is a family of shortened cyclic codes, whose codewords can always be computed working in T w. Fact 3.7. Let C(D) be a shortened cyclic code such that D = 1. Then Algorithm 3.5 needs computations only with coefficients in F q. Proof. It is well-known in Coding theory that shortening once a cyclic code gives rise to equivalent codes. So, if D = 1, we can always suppose D = {0}. But {0} is a cyclotomic set for any q. We provide an example of application of Fact 3.7. Example 3.8. We compute the number A 5 of minimum weight word for code Ċ = C(D), which is the shortened code (in any position) of C = BCH(63, 5). We apply Algorithm 3.5 and Fact 3.7. Ideal ĴĊ(5) is generated by the {p i,j (z i, z j )} 1 i<j 5 and z 1 +z 2 +z 3 +z 4 +z 5, z1 3 +z2 3 +z3 3 +z4 3 +z5, 3 z1, l z2, l 5 l=0 l=0 62 l=0 z l 3, 62 l=0 62 z4, l z5 l. l=0

6 The leading terms of the reduced Gröbner basis w.r.t. degrevlex are: z 1 z 2 2 z 3 z 28 2 z 29 3 z 12 2 z32 4 z 13 3 z32 4 z 26 2 z30 4 z2 5 z3 27z30 4 z2 5 z4 58z2 5 z2 2z56 4 z2 5 z4 60 z2 4z56 4 z3 3z56 4 z2 5 z3 5z56 4 z5 62 z2 10z46 4 z6 5 z3 11z46 4 z6 5 z2 4z54 4 z6 5 z 5 z4 54z6 5 z2 10z30 4 z30 5 z3 11z30 4 z30 5 z2 2z54 4 z30 5 z3 3z54 4 z30 5 The number of solutions is η[5] = η({0})[5] = , the permutations are 5! and so the minimum weight codewords are A 5 = /120 = 1740 (fully compliant to known results). The argument presented so far are a generalization and improvements of arguments presented in [22], as we now show. Definition 3.9 ([22]). Let C be an F q [n, k, d] cyclic code with a defining set S C = {h 1,..., h r }. Let w be an integer 1 w n. We denote by J C (w) the ideal in R w generated by: J C (w) =< {y 1 z h y wz h 1 w,..., y 1 z hr y wz hr w } {z1 n+1 z 1,..., zw n+1 z w } {y q 1 1 1,..., yw q 1 1} >. It is clear that, for any C and any w, ideal J C (w) is larger than ideal Ĵ C (w), so that the former contains some spurious solutions. However, in some cases the author in [22] is able to count the spurious solutions with combinatorial arguments, so that ideal J C (w) can be used to determine the distance by checking the smallest w s.t. J C (w) contains some proper solutions. Unfortunately, no equivalent of Lemma 3.14 appears in said paper, so that J C (w) did not look useful for computing the weight distribution. By Lemma 3.14 we can now subtract from V(J C (w)) the number of spurious solutions and divide the result by w!. This is an alternative approach to Algorithm 3.5, but it can only be used in those cases where formulae for counting spurious solutions are known and it is not clear how similar combinatorial formulae might be obtained for shortened codes. In this sense, our results so far are an improvement and a generalization on results in [22]. We now provide an example, the computation of the full weight distribution of a small binary cyclic code, where we use both methods. Example We compute the weight distribution of C = BCH(15, 5). As it is a narrow-sense BCH code, it is enough to compute A i for 5 i 7 (A n i = A i and its distance is at least 5 for the BCH bound). The computation of J C (5) had already been made in [22], where it was shown that V(J C (5)) = 5536 and the number of spurious solutions is By Applying Lemma 3.14, we immediately get A 5 = ! = = 18. To compute A 6, we apply Theorem 3.3. We construct ĴC(6) and we compute its Gröbner basis Ĝ(6), whose leading terms are 6

7 {z 1, z 2 2z 3, z 4 2, z 5 3, z 2 2z 6 4z 2 5, z 3 3z 6 4z 2 5, z 10 4 z 2 5, z 12 4, z 14 5, z 15 6 }, so that η[6] = and hence A 6 = 21600/720 = 30. To compute A 7, we apply Theorem 3.3. We construct ĴC(7) and compute Ĝ(7), whose leading terms are {z 1, z2 2, z3 3, z4 4 z 5, z4 12, z13 5, z14 6, z15 7 }, so that η[7] = and hence A 7 = 75600/5040 = 15. We can summarize our results using the known symmetry of the weight distribution for classical BCH codes: A i = 0 for 1 i 4 and 11 i 14, A 0 = 1, A 5 = 18, A 6 = 30, A 7 = 15, A 8 = 15, A 9 = 30, A 10 = 18, A 15 = 1. For some C we can use a result from [22] in our new context, but only when we compute the number of minimum codewords, i.e. A d (C). We summarize these ideas in the following remark. Remark Let J be the ideal defined exactly as ĴC(d) except that polynomials {p i,j } are not included in its basis. If C is binary cyclic and 1 S C, it is proved in [22] that J contains no spurious solutions. This means that J contains the {p i,j } even if they are not added to the basis. From a computational point of view this is important, because sometimes the computation of the Groebner basis starting from the full basis of ĴC(d) can be too expensive, while the computation starting only from the basis of J can be lighter. Intermediate approaches are possible, by adding only some of the {p i,j }. We provide a table with some comparisons (Table 1), from which it is clear that sometimes it is convenient to use an approach, sometimes another. In Table 1 the comparison is between J C (δ), J C (δ) (obtained from J C (δ) by adding polynomial p δ 1,δ ) and J C (δ) (obtained from J C (δ) by adding polynomials {p δ 2,δ 1, p δ 2,δ, p δ 1,δ }). The comparison is done for some BCH(n, δ) codes, with varying lengths and designed distances. Table 1: Comparison Between J C (δ), J C (δ) and J C (δ) n δ A δ time J C time J C time J C (δ) < 1 sec < 1 sec < 1 sec < 1 sec < 1 sec < 1 sec sec 21 sec 1 sec sec 17 sec 8 sec sec 7 sec 8 sec sec 1116 sec 1202 sec sec 3 sec 5 sec sec 1116 sec 219 sec sec 454 sec 642 sec 7

8 3.1 Proof of Theorem 3.3 Let c = (c 0,..., c n 1 ) C be a codeword of weight w. There are w non-zero components in C. We may collect their positions in set {µ 1,..., µ w } and their values in set T = {ν 1,..., ν w } in such a way that c µi = ν i, for any 1 i w. Thus c = (0,..., 0, ν }{{} 1, 0,..., 0, ν i, 0,..., 0, ν w µ 1 1 µ 1 µ i µw, 0,..., 0). }{{} n 1 µ w However, we are not imposing any order on set {µ 1,..., µ w } and µ 1 is not necessarily the smallest of the {µ i }. Let H be the standard parity-check matrix of C. Since c C, we must have Hc = 0, i.e. (1) (α h 1 ) µ 1 ν (α hw ) µw ν w = 0,..., (α hr ) µ 1 ν (α hr ) µw ν w = 0. Let I 1 R w = F q [z 1,..., z w, y 1,..., y w ] be the ideal generated by {y 1 z h y wz h 1 w,..., y 1 z hr y wz hr w }. Let ɛ = (α µ 1,..., α µw, ν 1,..., ν w ) (F Q ) 2w. From equations (1) we immediately get that ɛ is a solution of I 1 corresponding to codeword c. Since I 1 depends only on w and not on c, we have that any codeword of weight w corresponds to at least one solution of I 1. In other words, we can say that variety V(I 1 ) contains points corresponding to codewords of C with weight w. We call this kind of points proper solutions. However, there are other points in V(I 1 ), apart from the proper solutions, and we call them spurious solutions. First of all, since α F Q and T F q, we can discard all points outside F w Q Fw q. Second, since α is a (primitive) n-th root of unity, we have that all its powers satisfy z n = 1. This means that we lose no proper solutions if we consider the smaller variety V(I 2 ), where I 2 is the ideal in R w generated by I 1 and the two polynomial sets {y q 1 1 1,..., y q 1 w 1}, {z n 1 1,..., z n w 1}. Observe that condition y q 1 i = 1 ensures that the value lies in F q but also that it cannot be zero (the y variables correspond to error values and any error value cannot be zero). Although we have removed some spurious solutions, we still have some of them, because a point of type ɛ = (α µ 1,..., α µw, ν 1,..., ν w ), with µ i = µ j for some i j, cannot correspond to a codeword, since µ i and µ j must correspond to different positions for the error values ν i and ν j. To remove these spurious solutions, we need to add polynomials {p i,j } introduced in Definition 3.1, as we are going to show. 8

9 Definition Let K be any field. Let w and n be two integers such that 1 w n. Let E be the splitting field of x n 1 over K. We denote by V w,n the algebraic variety defined in E w as V w,n = {( z 1,..., z w ) z n i 1 = 0 for 1 i w, z i z j for 1 i j w}. We denote by I the vanishing ideal I = I K (V w,n ) Observe that V(I) = V w,n (by definition) and that points of variety V w,n correspond exactly to sets of w different n-th roots of unity. In the following fact, we give an explicit description for I in the polynomial ring K[z 1,..., z w ], imposing only condition (n, χ(k)) = 1, which is actually a standard condition of non-degeneracy in cyclic code theory (and which arises in this more general setting). Proposition Let K be any field. Let w and n be two integers such that 1 w n and (n, χ(k)) = 1. Then we can take the following set of polynomials as a basis for I: F = {z n 1 1,..., z n w 1, } {p i,j } 1 i<j w. Proof. Let E be the splitting field of x n 1 over K. Let z = ( z 1,..., z w ) be any point in V w,n and let z = ( z 1,..., z w ) be any element in E n lying outside V w,n. We have to show that 1. z satisfies all polynomials in F and 2. there is at least a polynomial in F that is not annihilated by z. First, we deal with case 1. It is obvious that z annihilates every polynomial in F of kind z n i 1, by definition of V w,n. If we take i and j such that 1 i < j w, we have only to show that p i,j ( z i, z j ) = 0. As z n i = 1 = z n j, we have 0 = ( z n i z n j ) = ( z i z j ) p i,j ( z i, z j ), but z V w,n implies ( z i z j ) 0 and so p i,j ( z i, z j ) = 0. We now deal with case 2. If z has at least a component z i which is not an n-th root of unity, then polynomial zi n 1 in F does not vanish in z. If z has all components which are n-th roots of unity, but it has two components, z i and z j, which have the same non-zero value a, then polynomial p i,j does not vanish in (a, a), since and (χ(k), n) = 1. n 1 p i,j (a, a) = a h a n 1 h = na n 1 h=0 9

10 Returning to the proof of Theorem 3.3, by applying Proposition 3.13 it is now clear that we can consider ideal I 3 in R w generated by I 2 and F. It is also clear that I 3 = ĴC(w). Its variety V = V(I 3 ) does not contain spurious solutions of the previous type and we claim that actually it contains only points coming from codewords of C of weight w. To prove our claim, we take an arbitrary point ɛ = ( z 1,..., z w, ȳ 1,..., ȳ w ) in V. Since z n i 1 I 3 for any i, we can write any z i as z i = α µ i, for some 0 µ i n 1. Since z i z j for any i j and 0 µ i n 1 for any i, we have that µ i µ j for any i j. We now consider vector c in (F q ) n c = (0,..., 0, ȳ }{{} 1, 0,..., 0, ȳ i, 0,..., 0, ȳ w µ 1 1 µ 1 µ i µw, 0,..., 0). }{{} n 1 µ w A direct check shows that c is actually a codeword (because it satisfies H c = 0) and hence our claim has been proved. We now know that variety V(ĴC(w)) contains only proper solutions and that to any codeword of weight w corresponds at least a point in V(ĴC(w)). We can show more in the following lemma. Lemma Let (ȳ, z) = (ȳ 1,..., ȳ w, z 1,..., z w ) be a solution of system Ĵ C (w), which corresponds to a codeword x C. Let π S w, where S w is the symmetric group on w elements. Then any permutation (ȳ, z) π of (ȳ, z) of type (ȳ π(1),..., ȳ π(w), z π(1),..., z π(w) ) is a solution of ĴC(w) and corresponds to the same codeword x. Furthermore, the {(ȳ, z) π } are distinct as π varies. Proof. System ĴC(w) is obviously invariant under variable permutations of the following nature: (y 1,..., y w, z 1,..., z w ) (y π(1),..., y π(w), z π(1),..., z π(w) ), where π S w, so that any such permutation of any solution of Ĵ C (w) is again a solution of ĴC(w). As variables {z j } correspond to the positions of the non-zero coordinates of the codeword x, their ordering is inessential. Similarly for the {y j }, which correspond to the values of these non-zero coordinates, provided they follow their associated {z j } variables in the permutation. It is then clear that the {(ȳ, z) π } are distinct as π varies. Let c be a codeword of weight w in C(D). By definition of shortening this codeword can be seen as a codeword in C of weight w, such that it has zero components at least in positions of D. Since c is a codeword in C of weight w, it corresponds to w! solutions of ĴC(w). Since all its components in positions of D are zero, we have that its corresponding solutions satisfy also p i (z i ) = 0. But the adding of polynomials {p i } to ĴC(w) is exactly what is needed to obtain ĴC(D)(w). 10

11 In conclusion, to any codeword of weight w in C(D) there correspond w! solutions of ĴC(D)(w). Moreover, all solutions of ĴC(D)(w) are of this kind. These two facts and Lemma 3.14 prove Theorem Conclusions In this paper a method to compute the weight distribution of shortened (and non-shortened) cyclic codes has been shown, along with applications and some special cases. Our method depends on the computation of a Gröbner basis for a suitable polynomial ideal. This approach was pioneered in [6], where a related ideal (called the CRHT ideal ) was proposed in order to decode cyclic codes up to their actual distance (rather than up to the BCH bound or other similar bounds). In fact, the ideal used in [4, 18, 22, 23] is a natural application of the CRHT ideal and the ideal proposed in Definition 3.2 is a direct improvement of the latter. Moreover, an improvement on the structure of the ideal CRHT itself has led to new decoding algorithms for cyclic codes ([21, 20]). However, this paper does not deal uniquely with cyclic codes, but extends these Gröbner basis applications to other classes of codes. We have some work in progress for classical Goppa codes and other algebraic codes. 5 Acknowledgments The author wishes to thank Patrick Fitzpatrick, Teo Mora and Carlo Traverso, for many fruitful suggestions and hints. He also wishes to thank the anonymous referees, whose comments greatly improved the paper presentation. A special thank to Marc Giusti, for the use of the computation resources of MEDICIS [17], and to all the team of Singular, which has been the preferred software package for computing Gröbner bases. This work has been partially supported by STMicroelectronics contract Complexity issues in algebraic Coding Theory and Cryptography. References [1] A.M. Barg, I.I. Dumer, On Computing the Weight Spectrum of Cyclic Codes, IEEE Trans. on Inf. Th., vol. 38, no. 4, p , Jul [2] E. Betti, M. Sala, A new bound for the minimum distance of a cyclic code from its defining set, IEEE Trans. on Inf. Th., vol. 52, no. 8, p , [3] A.M. Bigatti, P. Conti, L. Robbiano, C. Traverso, A Divide and Conquer algorithm for Hilbert-Poincaré Series, Multiplicity and Dimension 11

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