ISIT 2004, Chicago, USA, June 27 – July 2, 2004
A Generalization of MDS Codes1 Yuan Luo
Fang-Wei Fu
Chaichana Mitrpant, A. J. Han Vinck
Computer Science and Engineering Department Shanghai Jiao Tong University Shanghai 200030, China e-mail:
[email protected]
Temasek Laboratories National University of Singapore 10 Kent Ridge Crescent Singapore 119260 e-mail:
[email protected]
Institute for Experimental Mathematics Essen University Ellernstr. 29, 45326 Essen Germany e-mail:
[email protected] [email protected]
I. Introduction
Theorem 1 Let C 1 ⊇ C 2 ⊇ C 3 be linear codes of length n and dimension a1 ≥ a2 ≥ a3 , respectively. Then, (C 1 , C 3 ) is a relative MDS pair if and only if (C 1 , C 2 ) is a relative MDS a1 (C 2 , C 3 ) = a2 − a3 . pair and K
In an application of linear codes to cryptology, Ozarow and Wyner provided a noiseless wire-tap channel of type II with coset coding scheme in [3]. In [2], we considered a noiseless wire-tap channel of type II with non-legitimate parties (or side information). All parties are coordinated in coding their data symbols by using a same coset-coding encoder. The adversary has more power than that of [3] and [4], i.e. he can tap not only partial transmitted symbols but also the data symbols of non-legitimate parties. An inverse relative dimension/length profile (IRDLP) of a pair of codes was introduced to describe the equivocation to this more powerful adversary. The IRDLP is a generalization of the inverse dimension/length profile (IDLP) of Forney [1]. We derived a generalized Singleton bound on IRDLP in [2] to maximize the equivocation to the more powerful adversary. In this paper, the maximum distance separable (MDS) code is extended to a two-code format: relative MDS pair. For a code C 1 and a subcode C 2 , the pair (C 1 , C 2 ) is called a relative MDS pair if its IRDLP reaches the generalized Singleton bound. We consider properties and constructions of relative MDS pairs.
Theorem 2 Let C 1 be an [n, a1 ] linear code over GF (q) and C 2 be a subcode. Then, (C 1 , C 2 ) is a relative MDS pair if and only if Ai (C 1 ) = Ai (C 2 ) for 0 ≤ i ≤ n − a1 .
II. Main Results
Therefore, we can define a minimum relative MDS subcode of C 1 :
For a linear block code C over GF (q), let PJ (C) denote its projection code and let CJ denote {(c1 , · · · , cn ) ∈ C : ct = 0 for all t ∈ J}, where J ⊆ {1, 2, ..., n}. The inverse relative dimension/length profile (IRDLP) of a code C 1 and a subcode 1 , C 2 ) = {K i (C 1 , C 2 ) : 0 ≤ i ≤ n}, C 2 is a sequence K(C where
i (C 1 , C 2 ) = min{dim[PJ (C 1 )] − dim[PJ (C 2 )] : |J| = i}. K 2
If C is a zero code consisting of only one codeword (0, . . . , 0), 1 ) is retrieved, see Forney [1]. then K(C Lemma 1 ([2]) For an [n, a1 ] linear code C 1 over GF (q) and 1 , C 2 ) is a subcode C 2 with dimension a2 , their IRDLP K(C upper bounded by U P (K):
For a relative MDS pair (C 1 , C 2 ), Theorem 1 presents a relation among C 1 , C 2 and some other subcodes. It is useful in constructing more relative MDS pairs from known relative MDS pairs. Let {Ai (C) : 0 ≤ i ≤ n} denote the weight distribution of a linear code C of length n. We have,
Theorem 2 provides a way to determine a relative MDS pair (C 1 , C 2 ) by considering the weight distributions of “light weighted” codewords in C 1 and C 2 . By using Theorem 2, we have following results. Lemma 2 Suppose (C 1 , C 2 ) and (C 1 , C 3 ) are both relative MDS pairs. Let C 4 = C 2 ∩ C 3 . Then (C 1 , C 4 ) is also a relative MDS pair.
∆(C 1 ) = ∩{Λ : (C 1 , Λ) is a relative MDS pair}. Theorem 3 Let C 1 be a linear code and C 2 be a subcode. (C 1 , C 2 ) is a relative MDS pair if and only if ∆(C 1 ) ⊆ C 2 . Furthermore, a code C 1 is an MDS code if and only if ∆(C 1 ) is a zero code. Corollary 1 For an [n, a1 ] code C 1 over GF (q), the number of relative MDS pairs (C 1 , C 2 ) for all possible subcodes C 2 is
q a1 −m − q t
a1 −m h−1
h=0 t=0
1, C 2) = We call (C 1 , C 2 ) a relative MDS pair if K(C 1 2 U P (K), or equivalently Ka1 (C , C ) = a1 − a2 . If C 2 is a zero code, then the definition of a relative MDS pair (C 1 , C 2 ) is the definition of an MDS code C 1 . In addition, (C 1 , C 1 ) is always a relative MDS pair. 1 This work was supported by German Science Foundation-DFG, and National Natural Science Foundation of China (Grant no. 90104005, 60173032, 60303026).
,(((
,
where m is the dimension of ∆(C 1 ).
More details about the constructions of relative MDS pairs are in our full paper.
i : 0 ≤ i ≤ n} = {0, · · · , 0, 1, 2, · · · , a1 −a2 , · · · , a1 −a2 }, {U P (K) i = 0} = a2 . where max{i : U P (K)
qh − qt
References [1] G. D. Forney, “Dimension/Length profiles and trellis complexity of linear block codes,” IEEE Trans. Inform. Theory, vol. 40, no. 6, pp. 1741-1752, 1994. [2] Y. Luo, C. Mitrpant, A. J. Han Vinck and K. Chen, “Some new characters on the wire-tap channel of type II,” submitted to IEEE Trans. Inform. Theory. [3] L. H. Ozarow and A. D. Wyner, “Wire-tap channel II,” AT&T Bell Laboratories Technical Journal, vol. 63, no. 10, pp. 21352157, 1984. [4] V. K. Wei, “Generalized Hamming weights for linear codes,” IEEE Trans. Inform. Theory, vol. 37, no. 5 pp. 1412-1418, 1991.
