Construction of MDS Self-dual Codes over Finite Fields

07/27/2018 ∙ by Khawla Labad, et al. ∙ Central China Normal University 0

In this paper, we obtain some new results on the existence of MDS self-dual codes utilizing (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. For some fixed q, our results can produce more classes of MDS self-dual codes than previous works.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

A linear code over a finite field of length , dimension and minimum distance is called MDS (maximum distance separable) if it attains the Singleton bound: . MDS codes have been of much interest from many researchers due to their theoretical significant and practical implications, see [References], [References], [References].

For a linear code , we denote the dual of under Euclidean inner product by . The linear code is called self-dual if . Self-dual codes have attracted attention from coding theory, cryptograph and other fields. It has been found various applications in cryptography, in particular secret sharing scheme [References], [References], [References] and combinatorics [References].

More recently, the application of MDS codes renewed the interest in the construction of MDS self-dual codes, see [References], [References], [References], [References]. K. Guenda [References] constructed MDS Euclidean and Hermitian self-dual codes which are extended cyclic duadic codes or negacyclic codes. She also constructed Euclidean self-dual codes which are extended negacyclic codes.

Generalized Reed-Solomon () codes is a class of MDS code which has nice algebraic structure. It can be systematically constructed and has been found wide applications in practice. MDS self-dual codes through codes have been studied by L. Jin and C. Xing [References], where they constructed several classes of MDS self-dual codes through codes by choosing suitable parameters. In [References], H. Yan generalizes the technique in [References] and construct several classes of MDS self-dual codes via codes and extended codes.

Since MDS self-dual codes over finite field of even characteristic with any possible parameter have been found in [References]. In this paper, we obtain some new results on the existence of MDS self-dual codes through (extended) codes over finite fields of odd characteristic. Some results in this paper extend those of [References] and [References]. Comparing to previous works, for some fixed square prime power , our construction will produce more -ary MDS self-dual codes.

This paper is organized as follows. In Section 2 we will introduce some basis knowledge and auxiliary results on codes and extended codes. In particular, Corollary 2.1 and Corollary 2.2 give a criterion for an (extended) code to be self-dual. In Section 3 we will present our main results on the construction of MDS self-dual codes. Our main tools are Corollary 2.1 and Corollary 2.2. We choose suitable parameters to make the conditions in Corollary 2.1 and Corollary 2.2 hold.

00footnotetext: Some results in Table 1 are overlapped. For instance, columns 6 and 7 in [References] are special cases of column 4 in [References].
even Reference
even [References]
odd [References]
odd , [References]
odd , [References]
, prime and odd [References]
, , odd , odd and prime [References]
, odd, , even and [References]

, odd,
, even , and [References]

, odd,
, odd , and [References]
, odd, , odd , and [References]

[References]
for any [References]

, odd
, even and [References]

, odd
, odd and [References]

[References]
[References]

, odd prime
, [References]
, odd prime , , [References]
Table 1: Known results on MDS self-dual codes of length     ( is the quadratic character of )
(even) Reference
, odd , , even Theorem 1
, odd , odd, and Theorem 2
, odd , even, and Theorem 3
, odd prime , and , even Theorem 4
Table 2: Our results

2 Generalized Reed-Solomon codes

In this section, we introduce some basic notations and results on generalized Reed-Solomon code. Throughout this paper, let be a finite field with elements, and let be a positive integer with . Choose to be an -tuple of distinct elements of . Put with . For an integer with , then linear code

(1)

is the generalized Reed-Solomon or code.
The code has a generator matrix

It is well-known that the code is a -ary -MDS code and its dual is also MDS.

We define

(2)

The dual of code is explicitly determined. Precisely, let

be the all-one vector with appropriate length. The dual of

is , where with for .

Remark 2.1.

The matrix is a generator matrix of , and is a generator matrix of . It is clear that

where is the diagonal matrix with diagonal entries .

We now introduce some basic notations and results on extended generalized Reed-Solomon code. The -dimensional extended code of length given by

(3)

where is the coefficient of in .
Clearly, the code has a generator matrix

It is known that is -ary -MDS code and its dual is also MDS. Precisely, the dual code of is , where with for .

Remark 2.2.

Let be a generator matrix for , and let be a generator matrix for . It is easy to see that the relationship between , and is follows

where is the diagonal matrix with diagonal entries .

The following lemma gives a criterion for a code to be self-dual.

Lemma 2.1.

Let be an even integer, and . The code is a -ary MDS self-dual code over if and only if for any codeword , where , we have .

Proof.

Since , for any , we have , where is a generator matrix for . By remark(2.1), we can obtain that

This implies that is MDS self-dual if and only if

In other words, . Therefore, the lemma is proved. ∎

Corollary 2.1.

([References], Corollary 2.4) For an even integer , and . If there exist such that for some for all , then the code defined in (1) is MDS self-dual, where for all .   

The following lemma is important and it gives the necessary and sufficient condition to construct self-dual codes via the extended codes.

Lemma 2.2.

Let be even and . The code is a -ary MDS self-dual code over if and only if for any codeword , where , we have .

Proof.

Since is a -ary MDS self-dual code , we have , where is a generator matrix for , with . By remark(2.2), we get

It follows that is a -ary MDS self-dual code over if and only if

That means . This completes the proof.

Corollary 2.2.

([References], Lemma 2) Let be even and . If for some for all , then the code defined in (3) is MDS self-dual, where for all .   

3 Main Results

The existence of MDS self-dual codes over finite fields of even characteristic has been completely addressed in [References]. In this section, we construct serval new classes of MDS self-dual codes, by using codes and extended codes over finite fields of odd characteristic.
The following lemma can be found in [References].

Lemma 3.1.

Let be a positive integer and let be a primitive -th root of unity. Then for any , we have

Let denote the set of nonzero squares of .

Theorem 1.

Let where is odd prime power, and let be a positive integer with and . Assume both and are even. Then there exists a -ary MDS self-dual code over .

Proof.

Let be a primitive -th root of unity. Since one has group isomorphism and embedding

and , we can choose () to be cost representatives of and

Then the entries of are distinct elements of . Note that

for any . By Lemma 3.1, for any and for any , we have

Note that since .

Since is even and is primitive -th root of unity, then . As a consequence, . By Corollary 2.1, there exists a -ary MDS self-dual code over .

Remark 3.1.

Theorem 3.4 (i) in [References] is a special case of the preceding result when .

Example 3.1.

When and , we can choose and . In this case and the preceding result shows that there exists -ary MDS self-dual code of length which, as our best knowledge, has not been found in previous references.

Theorem 2.

Let where is odd prime power, and let be a positive integer with and . Assume is odd. Then there exists a -ary MDS self-dual code over .

Proof.

The proof is similar as that of Theorem 1.

Note that since .

Since is odd, then . Therefore . By Corollary 2.2, there exists a -ary MDS self-dual code over .

Theorem 3.

Let where is odd prime power, and let be a positive integer with and . Assume is even. Then there exists a -ary MDS self-dual code over .

Proof.

The elements and are chosen in the same way as in Theorem 1. We define the generalized Reed-Solomn code with

Similar as the above proofs, for any and for any ,

and

Note that , and . Hence both and are in . By Corollary 2.2, there exists a -ary MDS self-dual code of length . ∎

Theorem 4.

Let with odd prime. For any with and , if is even, then there exists self-dual MDS code with length .

Proof.

Let be an -dimensional -vector subspace in satisfying . Denote by a primitive element of order . Choose

For any ,

(4)

where the last equality follows from that and runs through when runs through .

Denote by . Then . Since is even, we can deduce that . As a consequence, which is independent of the choice of . In this case we choose . Then . By Corollary 2.1, there exists self-dual MDS code with length . ∎

Remark 3.2.

For , the preceding result is exactly Theorem 4 (ii) in [References].

In the case is a square, usually Theorems 1-4 will present more classes of MDS self-dual codes than the previous results.

Example 3.2.

For , there are 243 different for which MDS self-dual code of lengths are constructed in all the previous works(in Table 1). In our constructions (Theorems 1-4), there are 713 different classes of MDS self-dual codes of different lengths.

Acknowledgements

This work is supported by the self-determined research funds of CCNU from the colleges’ basic research and operation of MOE (Grant No. CCNU18TS028). The work of J.Luo is also supported by NSFC under Grant 11471008.

References

  • [1] T. Aaron Gulliver, J.L. Kim, and Y. Lee, New MDS and near- MDS self-dual codes, IEEE Trans. Inf. Theory, vol. 54, no. 9, pp. 4354–4360, Sept. 2008.
  • [2]
  • [3] K. Betsumiya, S. Georgiou, T. A. Gulliver, M. Harada, and C. Koukouvinos, On self-dual codes over some prime fields, Discrete Math., vol. 262, nos. 1-3, pp. 37–58, 2003.
  • [4]
  • [5] R. Cramer, V. Daza, I. Gracia, J. J. Urroz, G. Leander, J. Marti-Farre, C. Padro, On codes, matroids and secure multi-party computation from linear secret sharing schemes, IEEE Trans. Inf. Theory, vol. 54, no. 6, pp. 2647–2657, June 2008.
  • [6]
  • [7] S. T. Dougherty, S. Mesnager, and P. Sol, Secret-sharing schemes based on self-dual codes, in Proc. Inf. Theory Workshop, pp. 338–342, May 2008.
  • [8]
  • [9] S.H. Dau, W. Song, Z. Dong, and C. Yuen, “Balanced spareseset generator matrices for MDS codes, ” in Proc. Inter. Symp. Inf. Theory, July 2013, pp. 1889–1893.
  • [10]
  • [11] S. Georgiou and C. Koukouvinos, MDS self-dual codes over large prime fields, Finite Fields Their Appl., vol. 8, no. 4, pp. 455–470, Oct. 2002.
  • [12]
  • [13] M. Grassl and T. Aaron Gulliver, On self-dual MDS codes, Proceedings of ISIT, 2008, pp. 1954–1957.
  • [14]
  • [15] K. Guenda, New MDS self-dual codes over finite fields,Designs Codes Cryptogr., vol. 62, no. 1, pp. 31–42, Jan. 2012.
  • [16]
  • [17] L. Jin and C. Xing, New MDS self-dual codes from generalized Reed- Solomon codes, IEEE Trans. Inf. Theory, vol. 63, no. 3, pp. 1434–1438, Mar. 2017.
  • [18]
  • [19] J.I. Kokkala, D.S. Krotov and P.R.J. stergrd, “On the classification of MDS codes”, IEEE Trans. Inf. Theory, vol. 61, no. 2, pp. 6485–6492, Dec. 2015.
  • [20]
  • [21] E. Louidor and R.M. Roth “Lowest density MDS codes over extension alphabets”, IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 3187–3197, July 2006.
  • [22]
  • [23] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North Holland, 1977.
  • [24]
  • [25] J. Massey, Some applications of coding theory in cryptography, in Proc. 4th IMA Conf. Cryptogr. Coding, 1995, pp. 33–47.
  • [26]
  • [27] J. L. Kim and Y. Lee, Euclidean and Hermitian self-dual MDS codes over large finite fields, J. Combinat. Theory, Series A, vol. 105, no. 1, pp. 79–95, Jan. 2004.
  • [28]
  • [29] H. Yan, A note on the construction of MDS self-dual codes, Cryptogr. Commun., Published online, March 2018, see https://doi.org/10.1007/s12095-018-0288-3
  • [30]