A shortening of C on the coordinate i is the [n 1, k 1, d] linear code obtained by leaving only such codewords and deleting the ith coordinate.

Transcription

1 ENEE 739C: Advanced Topics in Signal Processing: Coding Theory Instructor: Alexander Barg Lecture 6a (draft; 9/21/03. Linear codes. Weights, supports, ranks. abarg/enee739c/course.html The goal of this lecture is to study general properties of linear codes. Let C be a q-ary [n, k] code ith eight distribution {A i, i 0, 1,..., n}. Let G, H be a generator and a parity-check matrix of C. The dual code C is the set C {x H n q : Hx T 0}. Let i be a not all-zero coordinate. We have {c C : c i 0} 1 q C A shortening of C on the coordinate i is the [n 1, k 1, d] linear code obtained by leaving only such codeords and deleting the ith coordinate. Let [n] : {1,..., n}, E [n], Ē [n]\e. For a matrix M ith n columns e denote by M(E the submatrix formed by all the columns ith numbers in E. The support of a vector x Hq n is defined as supp(x {i : x i 0}. Given a subcode A C, its support is supp(a {i [n] : a A a i 0} A shortening of C on the coordinates in Ē is a linear code CE such that x C E supp(x E. A projection of C on the coordinates in Ē, also called puncturing, is a restriction of C to the coordinates in E (simply said, deleting the coorinates in Ē from every codevector of C and deleting repeated entries from the result. The basic facts about shortenings and puncturings are given in the folloing Theorem 1. (i If t d(c 1, then C E is an [n t, k, d(c t] code. (ii dim C E E rk(h(e, dim(c E rk(g(e, dim(c E + dim((c E E. (iii C E C/CĒ. (iv (C E (C E ; (C E (C E. Proof : (i, (iii are obvious. (ii is counting the number of solutions of a system of linear equations. Let us prove the first part of (iv. Let a (C E, then G(Ea T 0, so a (C E. Further, by (ii The second part of (iv is analogous. dim(c E E rk(g(e E dim C E dim(c E. Recall the notation for the Gaussian binomial coefficient: [ ] k 1 n q n i 1 k q k i 1 In other ords, [ ] n [n] k [k][n k] u 1 here [u] (q u 1. Lemma 2. The number of linear [n, k] codes is [ n k]. Proof : [ ] n k equals the number of choices of k linearly independent vectors in H n q divided by the number of different bases of an [n, k] code. Lemma 3. Let E. Then (1 E rk(h(e k rk(g(ē 1

2 2 Proof : Let C E proj E C be the projection of C on the coordinates in E. Clearly, dim C E rk(g(e. On the other hand, C E C/CĒ by Theorem 1(iii; hence by (ii dim C E k dim CĒ k E + rk(h(ē. This lemma bridges linear codes ith the branch of combinatorics knon as matroid theory. It turns out that it is possible to abstract from the linear-algebraic setting and define dependencies axiomatically. One starts ith a finite set [n] and calls some of its subsets independent, making sure they satisfy the same set of natural conditions that the subsets of linearly independent vectors ould. Then for any subset E n one defines its rank as the size of a maximum independent subset contained in it. A large number of properties of independent sets and other subsets can be proved in this purely axiomatic setting, see [18]. Matroid theory finds uses in coding theory [4, 6], cryptography [5], information theory [10], multiuser communication [17] graph theory, topology [11]. Even though e ill not use this here, e ish to point out that this lemma contains the MacWilliams identities in disguise. Recall that ENEE722 gives a different proof of these identities, via the Fourier transform. Both proofs are due to F. J. MacWilliams [15]. Let n, i be nonnegative integers. Recall that n(n 1...(n i+1 i! i n 1 i 0 i 0 otherise In partilucar, ( 0 i δ0,i. Lemma 4. (The MacWilliams identities n u i u i (2 A i C q u A i. u n u Proof : We have the folloing chain of equalities: n u i (3 A i u E n u E n u (C E q n k u q n u rk(g(e E n u q u rk(h(ē u i q n k u A i. n u Here the first equality follos by counting in to ays the size of the set {(E, c : E n u and c (C E, t(c n u}, the second one is straightforard, the third one (the central step in the proof is implied by Lemma 3, and the final step follos by the same argument as the first one. A familiar form of these identities is obtained by using the eight enumerators of the code C and its dual code C. We have A(x, y 1 C A(x + (y, x y. The MacWilliams identities exist in a variety of different forms, see [13]. Remark. You should be aare that the linear-algebraic approach to the MacWilliams identities is only one of a number of possible points of vie. First of all, a substantial portion of the results above can be extended to nonlinear codes [16, Ch.5,6]. A combinatorial perspective is given in P. Delsarte [8], see also

3 3 [16, Ch. 21], [9]. An approach via harmonic analysis is given in G. Kabatiansky and V. Levenshtein [14], see also [7, Ch.9]. Binomial moments of the eight distribution. The quantities i C A i ( 0, 1,..., n n are called binomial moments of the eight distribution of the code C. We have seen (3 that C q dim(ce. E Clearly for an [n, k, d] code, C 0 1, C ( n, 1,..., d 1. It is possible to express the eight coefficients via the binomial moments. Lemma 5. Proof : A i i ( 1 i C. n i i ( 1 i C n i i j ( 1 i A j n i n i A j j0 i j0 A i j0 j ( ( n n j ( 1 i n i n A j ( n j n i j ( i j ( 1 i i Lemma 6. Let A(x, y be the eight enumerator of C and let C(x, y n C x n y. Proof : From (2 n C (x y n y A(x, y C(x y, y n n C y ( 1 n j x j y n j j j0 n n j x j y n j ( 1 n j C j j0 { } i n j n x n i y i A i. C n u q n k u C u n x j y n j j0 i ( 1 i n i (u 0,..., n. Introducing the eight enumerators C(x, y C i x n i y i, C (x, y C i xn i y i e then have C n u x n u y u q k C u (xq n u y u C C (y, x q k C(qx, y, a remarkably simple form of the MacWilliams equation.

4 4 An interesting observation about the binomial moments is that hile it is not possible to bound belo individual coefficients of the eight distribution, it is possible to do this for linear combinations of these coefficients (binomial moments. Here is one possibility: Theorem 7. Proof : Let E [n]. We have C dim(c E E rk(h(e q max(0, n+k q min(,n k q max(0, n+k This result can be applied to bounding the probability of undetected error P u of a linear [n, k] code. We have Theorem 8. [1] Let C be an [n, k] linear q-ary code. n ( p ( P u (q n+k 1 1 q n p. n k+1 Proof : From the previous theorem, Then C max(0, q n+k 1. p ( P u (C A(1 p, (1 pn C 1 p q, p n ( p ( C 1 q n p n ( p n ( p ( (C 1 q n p n ( p ( max(0, q n+k 1 1 q n p (1 p n ( 1 q p n In many cases the results of the last to theorems can be improved and generalized (to nonlinear codes, see [3]. The next exercise is the first step along this ay. Exercise. (Forget the 0 Try to define binomial moments of the distance distribution C for an unrestricted (i.e., not necessarily linear binary code C. Begin ith the folloing question: hat is the support of a subcode? (Hint: start ith subcodes of size 2. Prove that Lemma 6 still holds true, ith A(x, y replaced by the distance enumerator B(x, y. The rank function. Rerite (3 by collecting on the right-hand side subsets of one and the same rank. Namely, let U v u {E {1, 2,..., n} E u, rk(g(e v}. For instance, U k k (4 is the number of information sets of the code C. By (3 and Theorem 1(ii e have k n i A i q v (U v n, here the numbers U are the rank coefficients of C. Further, Lemma 3 implies that (5 (U u k+v u v0 U v n u. The last to equations relate the eight enumerator of C and its rank distribution (U v u, 0 u n, 0 v k.

5 5 Example. (MDS codes An (n, M, d code is called maximum distance separable (MDS if M q n d+1. Let C be an [n, k, n k+1] q-ary linear MDS code ith a parity-check matrix H. Then rk(h(e min{ E, n k} for all E S, 0 E n k. Therefore {( n (U v u u if (0 v u n k or (u n k + 1, v n k; 0 otherise. This enables us to compute the eight spectrum of C. Substituting the values of (U v u into (4, e obtain A 0 1, A i 0 for 1 i n k, and l 1 k + l A n k+l ( 1 j (q l j 1 (1 l k. k l j j0 Definition 1. The rank polynomial of a linear code C is U(x, y n u0 v0 k Uux v u y v. Relations of the rank polynomial of C, its dual code C, and the eight polynomial of C are given by the folloing results. Theorem 9. Proof : We have (6 (7 U (x, y ( 1 U (x, y x n y dim C U xy, y. n n k (U v ux u y v u0 v0 u0 v0 n n k Un u k u+v x u y v (by (5 u0 v0 n n k x n Uu k n+u+v x u y v n n k x n y n k Uu u v (xy u y u v x n y n k n u0 v0 u u0 vu n+k U v u(xy u y v. No let E n u, rk(g(ē v. Then Ē u and by (1 0 rk(h(e E k + v v (u n + k. Hence, for 0 v u n + k 1 e have U v u 0. Thus, the summation interval of v in (7 can be extended to 0 v u. Theorem 10. ( x y A(x, y y n C U, 1 y q (x y n U ( qy x y, 1. q Information profile of a linear code. Let C be a linear code, [n] {1,..., n}, E [n]. Consider the average (over the choice of E dimension of the code C E : ( 1 n e dim(c E. E ( [n] The number e can be thought of an the average information content of a -subset.

6 6 The numbers (e, 1,..., n constitute the information profile of the code C. Since dim(c E rk(g(e, here G is the generator matrix of C, e can also rite e iu i (C. The information profile as introduced for its one sake in [12] and as used recently in analyzing iterative decoding. Paper [12] also computes the average value of e over all [n, k] codes. Minimal vectors in linear codes. For the vectors x, y Hq n e rite x y (x y if supp(x supp(y (resp., supp(x supp(y. Definition 2. Let C Hq n be a linear code. A nonzero vector c C is called minimal if 0 c c implies c ac here c is another codevector and a is a constant. In ords: nonminimal vector covers a nonzero vector ith a smaller support. Let H be a parity-check matrix of C and let M be the set of its minimal vectors. Let us list basic properties of minimal vectors. Lemma 11. (i Let c C, U supp(c. Then c M if and only if rk(h(u U 1. (ii (U is minimal ( U n k + 1. (iii Every support of size U d(1 + 1 q 1 1 is minimal. (iv The linear span (the set of all linear combinations of M(C coincides ith C. Moreover, if q 2 then every nonzero codevector can be decomposed into a sum of minimal vectors ith pairise disjoint supports. (v Let C be a binary code. Then if c C, c M(C, then there is a pair of nonzero code vectors c 1 c and c 2 c ith disjoint supports such that c c 1 + c 2. Proof : The only if part of (1 is obvious. Let us prove the converse. Let h i be the ith column of H(U. By assumption, there exist U nonzero numbers λ i such that λ i h i 0 i1 and some 1 of these columns, say the first, are linearly independent. Suppose there exists a code vector c, c c, i.e., there exists a vanishing linear combination of columns that does not involve at least one of the first 1 columns, for instance, µ i h i 0 i2 ith µ 0. Multiply this sum by λ /µ and subtract from the first one. This gives a linear dependence beteen the first 1 columns, a contradiction. Part (ii is implied by (i. To prove part (iii, suppose that c C is a nonminimal vector of eight t(c d ( 1 + q and let c c, c C\{0}. Consider code vectors c ac, here a runs over all nonzero constants. Summing up their eights, e get ( t(c t(c. Thus, their average eight is t(c ( 1 t(c. One of these vectors, say c has eight at most the average. Together ith our assumption this implies a contradiction: t(c t(c t(c d( d 1 d 1. Part (iv Let c C\{0}. Suppose it is not minimal, then there is a minimal vector m 1 such that supp(m 1 supp(c. There alays is a nonzero constant a 1 such that (c a 1 m 1 c. Denote c 1 c a 1 m 1. By the same argument, either c 1 is minimal or there is an m 2 M such that (c 1 a 2 m 2 c 1. Since the

7 7 size of the support falls by (at least one in every step, for a certain i, the vector c i ill be minimal. Then c i c i 1 j1 a jc j. Part (v is obvious. Ho many minimal vectors are there in a typical linear code? Consider the ensemble given by random (n k n parity-check matrices. Let EM be the expected number of minimal vectors of eight. Theorem 12. [2] Let n k + 1. Then ( 2 EM q n k (1 q (n k i. VarM EM (1 + 2 d/2 EM. For the proof consult the source. In particular, this theorem implies that for n almost all vectors of eight n k + 1 in a typical linear code are minimal. The main uses of minimal vectors are in decoding of linear codes and cryptography (access structures in secret sharing schemes. Example: Let C be an [n qm 1 q 1, n m 1, 3] Hamming code. We have M 1 2 (q m q i, (3 m + 1.! Proof: Consider s 1 linearly independent columns in the parity-check matrix H of the code C. The total number of linear combinations of these columns ith nonzero coefficients equals ( s ; the 1/(q 1th fraction of them appear as columns in H distinct from the chosen columns (since they are linearly independent. Every choice of linearly dependent columns of hich s 1 are linearly independent, defines a minimal codeord. Thus, one has to count the number of distinct choices of s linearly independent columns in H. This number equals 1 ( s! n(n 1 n q2 1 (... n qs 1 1 Taking into account that all the ( 1 choices of 1 linearly independent columns ithin a given support of size yield one and the same codeord, e find that the number of minimal codeords of eight in the code equals 1 ( B ( 1! n(n 1 n q2 1 (... n qs 1 1 ( s, The substitution of the value of n gives the desired result. References 1. K. A. S. Abdel-Ghaffar, A loer bound on the undetected error probability and strictly optimal codes, IEEE Trans. Inform. Theory (1997, no. 5, A. Ashikhmin and A. Barg, Minimal vectors in linear codes, IEEE Trans. Inform. Theory 44 (1998, no. 5, , Binomial moments of the distance distribution: Bounds and applications, IEEE Trans. Inform. Theory 45 (1999, no. 2, A. Barg, The matroid of supports of a linear code, Appl. Alg. Engrg. Commun. Comput. 8 (1997, E. F. Brickell and D. M. Davenport, On the classification of ideal secret sharing schemes, J. Cryptology 4 (1991, T Britz, MacWilliams identities and matroid polynomials, Electron. J. Combin. 9 (2002, #R J. H. Conay and N. J. A. Sloane, Sphere packings, lattices and groups, Springer-Verlag, Ne York-Berlin, P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Research Repts Suppl. 10 (1973, P. Delsarte and V. I. Levenshtein, Association schemes and coding theory, IEEE Trans. Inform. Theory 44 (1998, no. 6, S. Fujishige, Entropy functions and polymatroids combinatorial structures in information theory, Electron. Comm. Japan 61 (1978, no. 4, I. M. Gel fand and R. D. MacPherson, A combinatorial formula for the Pontrjagin classes, Bull. Amer. Math. Soc. (N.S. 26 (1992, no. 2,

8 8 12. T. Helleseth, T. Kløve, and V. I. Levenshtein, On the information function of an error-correcting code, IEEE Trans. Inform. Theory 43 (1997, no. 2, W. C. Huffman and V. Pless, Fundamentals of error-correcting codes, Ne York: Cambridge University Press, G. Kabatyansky and V. I. Levenshtein, Bounds for packings on the sphere and in the space, Problems of Information Transmission 14 (1978, no. 1, F. J. MacWilliams, A theorem in the distribution of eights in a systematic code, Bell Syst. Techn. Journ. 42 (1963, F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes, 3 ed., North-Holland, Amsterdam, D. N. C. Tse and S. V. Hanly, Multiaccess fading channels. I. Polymatroid structure, optimal resource allocation and throughput capacities, IEEE Trans. Inform. Theory 44 (1998, D. J. A. Welsh, Matroid theory, Academic Press, London, 1976.