A class of linear partition error control codes inγ-metric

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1 World Applied Sciences Journal (Special Issue of Applied Math): 3-35, 3 ISSN c IDOSI Publications, 3 DOI: 589/idosiwasjam53 A class of linear partition error control codes inγ-metric Sapna Jain Department of Mathematics, University of Delhi, Delhi 7, India sapnajain@gmxcom Abstract: Linear partition error control codes in the γ-metric is a natural generalization of error control codes endowed with the Rosenbloom-Tsfasman(RT) metric [4] to block coding has applications in different area of combinatorial/discrete mathematics, eg in the theory of uniform distribution, experimental designs, cryptography etc In this paper, we formulate the concept of linear partition codes in theγ-metric derive results for the rom block error detection rom bock error correction capabilities of these codes Key words: Linear codes, RT-metric, error-block code INTRODUCTION K Feng LXu FJHickernesll [] initiated the concept of linear partition block code endowed with π-metric which is a natural generalization of the Hamming-metric codes Also, we know that the Rosenbloom-Tsfasman metric (or RT-metric or ρ-metric) is stronger than the Hamming metric [,5] Motivated by the idea to have linear partition block code endowed with a metric generalizing the RT-metric, we formulate the concept of linear partition codes equipped with block ρ-metric name this new metric as the γ-metric We derive the basic results for linear partition codes in the γ- metric including various upper lower bounds on their parameters study their rom block error detection block error correction capabilities in Section 3 Section 4 of this paper DEFINITIONS AND NOTATIONS Let q,n be positive integers with q = p m, a power of a prime number p Let F q be the finite field having q elements A partition P of the positive integer n is defined as: P : n = n +n +n s where The partition P is denoted as In the case, when n n n s,s P : n = [n ][n ] [n s ] P : n = [n ] [n ][n ] [n ] }{{}}{{} r - copies r - copies we write where [n t ] [n t ], }{{} r t- copies P : n = [n ] r [n ] r [n t ] rt n < n < < n t Further, given a partition P : n = [n ][n ] [n s ] of a positive integer n, the linear space F n q over F q can be viewed as the direct sum or equivalently F n q = F n q F n q F ns q, V = V V V s, wherev = F n q V i = F ni q for all i i s Consequently, each vectorv F n q can be uniquely written as a v = (v,v,,v s ) where v i V i = F q n i for all i s Herev i is called thei th block of block size n i of the vectorv Definition Let v = (v,v,,v s ) F n q = F n q F n q F ns q be ans-block vector of lengthnoverf q corresponding to the partition P : n = [n ][n ] [n s ] of n We define the γ-weight of the block vector v as w (P) γ (v) = max s {i v i } The γ-distance d (P) γ (u, v) between two s-block vectors of lengthnviz u = (u,u,,u s ) 3

2 v = (v,v,,v s ),u i,v i F ni q ( i s) corresponding to the partition P is defined as d γ (P) (u,v) = w γ (P) (u v) s = max {i u i v i } Thend γ (P) (u,v) is a metric onf n q = Fn q Fq n F ns q Note Once the partition P is specified, we will denote the γ-weight w γ (P) by w γ (v) γ-distance d γ (P) by d γ respectively Definition A linear partition γ-code (or lpγ-code) V of length n corresponding to the partition P : n = [n ][n ] [n s ], n n n s is a F q -linear subspace off n q = Fn q Fn q Fns q equipped with the γ-metric is denoted as [n,k,d γ ;P] code where k = dim Fq (V)d γ = dγ(v) = minimumγ-distance of the codev Remark 3 For P : n = [] n, the γ-metric (or γ-weight) reduces to the ρ-metric (or ρ-weight) respectively [4] For a partitionp : n = [n ][n ] [n s ] of the positive integer n, the γ-distance (or γ-weight) is always greater than or equal to the π-distance (or π-weight) [] respectively, ie of n i column vectors of length k each H i = (H (i),h(i),,h(i) n i ) is thei th block ofh of block size n i consisting ofn i column vectors of length(n k) each Definition 6 A set of blocks {H ii,h i,,h ir } {H,H,,H s } of the parity check matrixh is said to be linearly independent if the union of all column vectors in the blocksh ii,h i,,h ir is a linearly independent set over F q Otherwise, we say that the set of blocks {H ii,h i,,h ir } is linearly dependent Equivalently, we can say that a set of blocks{h ii,h i,, H ir } {H,H,,H s } is linearly independent over F q iff α i H i +α i H i + +α ir H ir = α i = α i = α ir =, where for all j r, α ij = (α (ij),α (ij),,α n (ij) ij ) F ni j q α ij H ij = α (ij) H (ij) +α (ij) H (ij) + +α (ij) n ij H n (ij) ij 3 SOME PROPERTIES OFlpγ-CODES We begin by stating three results for lpγ-codes without proof as the proof is straightforward π-metric π-weight γ-metric γ-weight Theorem 7 The minimum γ-weight minimum γ- distance of an lpγ-code V coincide QED Theorem 8 Example 4 Let n = q = 5 Let P : 5 = [][][] be a partition of n = 5 Then F5 5 can be viewed as F5 5 = F5 F 5 F 5 s = 3 Let v = (v,v,v 3 ) = ( ) Then wγ(v) = Similarly if u = (u,u,u 3 ) = () x = (x,x,x 3 ) = ( ), thenwγ(u) = wγ(x) = 3 respectively QED Definition 5 The generator parity check matrix of an [n,k,d;p] lpγ-code over F q where P : n = [n ][n ] [n s ], n n n s are given as G = [G,G,,G s ] H = [H,H,,H s ], where for all i s,g i = (G (i),g(i),,g(i) n i ) is the i th block of G of block size n i consisting (a) An lpγ-code detects all block errors of γ-weight t or less iff the minimum γ-distance of the code is at least t+ (b) An lpγ-code V corrects all block errors of γ-weight t or less iff the minimumγ-distance of the codev is at least t+ QED Theorem 9 LetV be an[n,k;p]lpγ-code overf q corresponding to the partition P : n = [n ][n ] [n s ] of n The minimum γ-distance of the code V is d iff first (d ) blocks of the parity check matrix H are linearly independent first d blocks of H are linearly dependent overf q QED Example Let n = 6, n k = 5 q = 3 Let P : n = 6 = [][][][3] be a partition of n = 6 Then s = 4 n =,n =,n 3 = n 4 = 3 Let H = (H H H3 H4 ) be the parity check matrix of a 3

3 [6,;P]lpγ-codeV as given below Let H = α H +α H +α 3 H 3 + +α 4 H 4 =, () where α = (α () ) F 3,α = (α () ) F 3,α 3 = (α (3) ) F 3 α 4 = (α (4),α(4),α(4) 3 ) F3 3 Thenα H +α H +α 3 H 3 +α 4 H 4 = implies α () +α() This gives +α (3) +α (4) = +α(4) +α(4) 3 α () =, α () =, α (3) = α (4) = α (4) = α (4) 3 Therefore, the only solutions of () areα = α = α 3 = () α 4 = (a,a,a) wherea F 3 Thus, the first three blocks of H are linearly independent the first four blocks of H are linearly dependent Hence the [6, : P] lpγ-code V with H as the parity check matrix has minimumγ-distance equal to 4 Theorem [Singleton s Bound] If V is an [n,k,d : P] lpγ-code over F q corresponding to the partition P : n = [n ][n ] [n s ], n n n s Then n +n + +n d n k () Proof By Theorem 9, the columns of first(d ) blocks of the parity check matrix H of the code V are linearly independent Since the number of rows in H is n k, equation () follows QED Definition An [n,k,d : P] lpγ-code over F q with P : n = [n ][n ] [n s ] is said to be maximum γ- distance separable (M γds) if equality holds in () ie if (n k) equals the sum of block sizes of first (d ) blocks Example 3 Let q = 5,n = Let P : = [][] be a partition of n = The [,;P] lpγ-codev with parity check matrixh = ( ) overf 5 is anmγds code with maximumγ-distance equal to We now obtain Hamming sphere packing bound for lpγ-codes For this, we first prove a lemma Lemma 4 If V (n,n,,ns) t,q denote the number of all s-block vectors of length n over F q of γ-weight t or less corresponding to the partition P : n = [n ][n ] [n s ], n n n s, then = + V (n,n,,ns) t,q t q n+n+ +nr r= (q nr ) (3) Proof Let u = (u,u,,u s ) F n q = F n q F n F ns q To make the γ-weight of u to be equal to r( r t), we have q nj choices for the j th block ( j r ) (q nr ) choices for the r th block only one choice viz zero for the l th block (r + l s) Therefore, the number of s-block vectors of length n of γ-weight r is given by q A (n,n,,ns) r,q = q n+n+ +nr (q nr ) (4) The result now follows by taking summation of (4) for r = to t adding to the resultant corresponding to the null vector QED Theorem 5 (Hamming Sphere Bound) Let V be an [n,k,d : P] lpγ-code overf q corresponding to the partition P : n = [n ][n ] [n s ], n n 3

4 n s Then q n k V (n,n,,ns) [d ]/,q, (5) where V (n,n,,ns) [d ]/,q is given by (3) [x] denotes the largest integer less than or equal tox Proof The proof follows from the fact that all the s- block vectors of length n = s n i γ-weight [d ]/ or less must belong to distinct cosets of the stard array the number of available cosets is q n k QED 4 GILBERT AND VARSHAMOV BOUNDS FOR lpγ-codes In this section, we obtain Gilbert bound, Varshmov bound a bound for rom block error correction in lpγ-codes We derive Gilbert bound first Theorem 6 (Gilbert bound) Letn,k,q be positive integers where q = p m (p prime) k n Let P : n = [n ][n ] [n s ], n n n s be a partition of n Let d be a positive integer satisfying d s Then there exists an [n,k;p] lpγ-code over F q with minimumγ-distance at least d provided ( ) n k log q V (n,n,,ns) d,q, (6) where V (n,n,,ns) d,q is given by (3) Proof We shall show that if (6) holds then their exists an (n k) n matrixh overf q such that no linear combination of(d ) or fewer blocks ofh is zero We define an algorithm for finding the blocks H,H,,H s of H whereh i = (H (i),h(i),,h(i) n i ) for all i s From the set of allq n k column vectors of length(n k) overf q, we choose blocks of columns of the parity check matrixh as follows: () The n column vectors in the first block H can be any vectors chosen from the set of q n k column vectors of lengthn k overf q satisfying where λ H, λ = (λ (),λ(),,λ() n ) F n q () The second block H = (H (),H(),,H() n ) can be any set ofn column vectors of length(n k) satisfying λ H +λ H, where for i, λ i = (λ (i),λ(i),,λ(i) n i ) F ni q, w γ (λ,λ ) = max {i λ i } d (j) Thej th blockh j = (H (j),h(j),,h(j) n j ) can be any set of n j column vectors of length(n k) satisfying λ H +λ H + +λ j H j (7) where for i j, λ i = (λ (i),λ(i),,λ(i) n i ) F ni q, w γ (λ,λ,,λ j ) j = max {i λ i) } d (8) (s) Thes th blockh s = (H (s),h(s),,h(s) n s ) can be any set ofn s column vectors satisfying where λ H +λ H + +λ s H s λ i = (λ (i),λ(i),,λ(i) n i ) F ni q for all i s, w γ (λ,λ,,λ s ) = s max {i λ i) } d If we carry out this algorithm to completion, then, H,H,,H s are the blocks of size n,n,,n s respectively of an(n k) n wheren = n i ( s ) block 33

5 matrix H such that no linear combination of blocks of H of γ-weight (d ) or less is zero meaning thereby that this matrix is the parity check matrix of anlpγ-code with minimum γ-distance at least d We show that the construction can indeed be completed Let j be an integer such that j s assume that the blocks H,H,,H j have been chosen Then the blockh j can be added to H provided (7) is satisfied The number of distinct linear combinations in (7) satisfying (8) including the pattern of all zeros is given by V (n,,nj) d,q wherev (n,,nj) d,q is given by (3) As long as the set of all linear combinations occuring in (7) satisfying (8) is less than or equal to the total number of (n k)-tuples, the j th block H j can be added to H Therfore, the blockh j can be added to H provided that or q n k V (n,,nj) d,q ( ) n k log q V (n,,nj) d,q Thus the fact that the blocksh,h,,h s can be chosen follows by induction onj we get (6) QED Corollary 7 Let P : n = [n ][n ] [n 3 ] be the partition of a positive integer n Let t be a positive integer satisfyingt+ s Then, a sufficient condition for the existence of an [n,k;p] lpγ-code over F q that corrects all rom block errors ofγ-weighttor less is given by ( ) n k log q V (n,,ns) t,q Proof The proof follows from Theorem 6 the fact that to correct all errors of γ weight t or less, the minimum γ-distance of an lpγ-code must be at least t+ QED Example 8 Let n = 3,k =,d = q = 5 Let P : 3 = [][] be a partition of n = 3 We show that for these values of the parameters, equation (6) is satisfied We note that heren =,n = Equation (6) for these parameters becomes or which is true 5 3 V (,3),5, 5 5 (sincev (,3),5 = 5), Therefore, by Theorem 6, there exists a [3,;P] lpγcodev overf 5 with minimumγ-distance at least Consider the following 3 block parity check matrix H of a [3,;P] lpγ-code V over F 5 constructed by the algorithm discussed in Theorem 6: H = 3 3 We claim that thelpγ-code which is the null space of the matrixh has minimumγ- distance at least The generator matrix of the lpγ-code corresponding to the parity check matrixh is given by G = [ 3 ] 3 = [3 ] 3 The five codewords of thelpγ-codev withgas generator matrix H as parity check matrix are given by: v = ( );wγ (v ) =, v = (3);w γ (v ) =, v = ( 4);wγ (v ) =, v 3 = (43);w γ (v 3 ) =, v 4 = ( 34);w γ (v 4 ) = Therfore, the minimum γ- weight of the lpγ-code V is equal to Hence Theorem 6 is verified Theorem 9 (Varshamov Bound) Let B q (n,d;p) denote the largest number of code vectors in an [n,k;p] lpγ-codev over F q with P : n = [n ][n ] [n s ] having minimumγ-distance at leastd Then B q (n,d;p) q n logq(l), where L = V (n,,ns) d,q is given by (3) x denotes the smallest integer greater than or equal to x Proof By Theorem 6, there exists an[n,k;p]lpγ-code overf q with minimumγ-distance at least d provided q n k V (n,,ns) d,q n k log q (L) k n log q (L) = L The largest integer k satisfying the above inequality is n log q (L) Thus B q (n,d;p) q n logq(l) 34

6 wherel = V (n,,ns) d,q is given by (3) QED Acknowledgment The author would like to thank her spouse Dr Arihant Jain for his constant support encouragement for pursuing this research work References Dougherty, ST MM Skriganov, MacWilliams duality the Rosenbloom- Tsfasman metric, Moscow Mathematical Journal, :83-99 Feng, K, L Xu F Hickernell, 6 Linear Error-Block Codes, Finite Fields Applications, : MacWilliams, FJ NJA Sloane, 977 The Theory of error Correcting Codes, North Holl Publishing Co 4 Rosenbloom, M Yu MA Tsfasman, 997 Codes for m-metric, Problems of Information Transmission, 33: Skriganov, MM, Coding Theory Uniform Distributions, St Petersburg Math J, 3: Udomkavanich, P S Jitman, Bounds Modifications on Linear Error-block Codes, International Mathematical Forum, 5: