Submitted papers

39. Quasi-cyclic self-dual codes based on building-up constructions,
with Sunghyu Han, Heisook Lee, and Yoonjin Lee, submitted 6/18/2009

Abstract: Ling and Sole proved that there is a one to one correspondence between l-quasi-

cyclic self-dual codes over Fq of length m and self-dual codes over R of length

lm, where R = Fq[Y ]=(Y^m - 1) and Fq is a finite field of order q. There are still

only few systematic methods known for constructing quasi-cyclic self-dual codes

over Fq. The main result of this paper is proving that every l-quasi-cyclic self-

dual code of length ml over Fq can be obtained by the building-up construction,

provided that char (Fq) = 2 or q = 1 (mod 4), m is a prime p, and q is a primitive

element of Fp. We also present a complete classification of all binary l-quasi-cyclic

self-dual codes of even length 5l <= 30, and we find optimal such codes of lengths

40, 50, and 60. Furthermore, when m = 5, we obtain a new 8-quasi-cyclic self-dual

[40, 20, 12] code over F3, and when m = 7, we find a new 4-quasi-cyclic self-dual

[28, 14, 9] code over F4 and a new 6-quasi-cyclic self-dual [42, 21, 12] code over F4.

 

38. Cubic self-dual codes based on building-up constructions,
with Sunghyu Han, Heisook Lee, and Yoonjin Lee
submitted, 1/12/2009

 

Abstract: In this paper we present a very efficient construction method for finding quasi-cyclic self-dual codes over finite fields. By using our building-up construction, we classify all binary cubic self-dual codes of length up to 24 and find a new quasi-cyclic self-dual [24,12,9] code over F_5, whose corresponding lattice by Construction A is shown to be the odd Leech lattice O_24.

 

37. Self-dual codes over Frobenius rings,
with Steven T. Dougherty, Hamid Kulosman, and Hongwei Liu
submitted, 12/31/2008, resubmitted 7/31/2009

 

Abstract: We prove that self-dual codes exist over all finite commutative Frobenius rings, via

their decomposition by the Chinese remainder theorem into local rings. We construct

non-free self-dual codes under some condition using self-dual codes over finite fields and we construct free self-

dual codes by lifting elements from the base finite field. We generalize the building-up

construction for finite commutative Frobenius rings, showing that all self-dual codes

with minimum weight greater than 2 can be obtained in this manner in cases where the construction applies.

 

36. Formally self-dual additive codes over F_4,
with Sunghyu Han
submitted, 11/05/2008, resubmitted 7/14/2009

(File for all formally self-dual (6,2^6,3) additive codes over GF(4))

(File for formally self-dual (9,2^9,4) additive codes over GF(4))

 

Abstract: Additive codes over F_4 have been of great

interest due to their application to quantum error-correction. As another

application, we introduce a new class of formally self-dual additive

codes over F_4, which is a natural analogue of binary

formally self-dual codes and is missing in the study of additive

codes over F_4. In fact, Gulliver and Ostergard

(2003) considered formally self-dual linear codes over F_4

of even lengths, and Choie and Sole (2008) suggested to classify

formally self-dual linear codes over F_4 of odd lengths in

order to study lattices from these codes. These motivate our study

on formally self-dual additive codes over F_4. In this

paper, we define extremal and near-extremal formally self-dual

additive codes over F_4 and classify or construct all

extremal and near-extremal codes. We discuss a general method

(called the weak balance principle) to construct such codes. We

conclude with some open problems.

 

35. Building-up constructions for self-dual codes,
with Yoonjin Lee
submitted, 6/2/2008, resubmitted, 12/19/2008

(File for new self-dual [32,16,9] codes over GF(3) with trivial automorphsim groups)

 

Abstract: The building-up construction for self-dual codes was developed

by the authors and H. Lee over finite fields GF(q) when q is a

power of 2 or q = 1 (mod 4), and then over finite rings Z_p^m with

p =1 (mod 4), and over Galois rings GR(p^m, r) with p = 1 (mod 4)

with any r or p = 3 (mod 4) with even r, where m is any positive

integer. In this paper, we complete the building-up construction for self dual

codes over GF(q) with q = 3 (mod 4), and over Z_p^m and Galois

rings GR(p^m, r) with an odd prime p satisfying p =3 (mod 4). We also

present a building-up construction for self-dual codes over finite chain

rings and p-adic rings Zp^∞ with any odd prime p. For example,

we construct new 945 extremal self-dual ternary [32,16,9] codes,

each of which has a trivial automorphism group.We also construct many self-dual

codes over Z_9 of lengths 12, 16, 20 all with minimum Hamming weight

6, which is the best possible minimum Hamming weight among free

self-dual codes over Z_9 of these lengths.

34. Triple circulant codes based on quadratic residues,
with Sunghyu Han
submitted, Mar. 14, 2008

Abstract: One of the most exciting classes of algebraic codes is the class of quadratic residue

(QR) codes over a finite field. A natural construction doubling the lengths of QR codes

seems to be the double circulant constructions based on quadratic residues given by

Karlin, Pless, Gaborit, et. al. In this paper we define a class of triple circulant linear

codes based on quadratic residues. We construct many new optimal codes or codes

with the highest known parameters using this construction. In particular, we find

the first example of a ternary [58, 20, 20] code, which improves the previously known

highest minimum distance of any ternary [58, 20] codes.

33. Constructions of self-dual codes over chain rings,
with Steven T. Dougherty and Hongweig Liu,
submission (6/20/2009) (file for generator matrices )

Abstract: We study self-dual codes over chain rings. We describe a technique for constructing

new self-dual codes from existing codes and we prove that for certain rings all self-dual

codes can be constructed by this technique. We extend this construction to self-dual

codes over principal ideal rings via the Chinese Remainder Theorem. We use torsion

codes to describe the structure of self-dual codes and to set bounds on their minimum

Hamming weight. Interestingly, we find the first examples of MDS self-dual codes of

lengths 6 and 8 and near-MDS self-dual codes of length 10 over a certain chain ring

which is not a Galois ring.

Accepted papers (in press)

32. Identifying codes in q-ary hypercubes,
with Seog-Jin Kim (resubmitted 4/17/2009, accepted 4/22/2009) To appear in Bulletin of the Institute of Combinatorics and its Application.

31. A generalized Gleason-Pierce-Ward theorem,
with Xiaoyu Liu
(submitted 7/23/2008, resubmitted 12/01/2008, accepted 4/1/09) To appear in Designs, Codes, and Cryptography.

30. Type I codes over GF(4),
with Hyun Kwang Kim and Dae Kyu Kim
To appear in Ars Combinatoria, (accepted on May 17, 2007)

Printed Journal Papers (contents in journals may be modified)

29. The nonexistence of near-extremal formally self-dual codes,
with Sunghyu Han
Designs, Codes, and Cryptography (second submission 8/21/08,

Accepted 9/29/2008, (5/21/07 original submission)), Vol. 51 (2009), No.1, pp. 69-77.

28.MDS codes over finite principal ideal rings,
with Steven T. Dougherty and Hamid Kulosman,
Designs, Codes, and Cryptography (submitted, Mar. 5, 2007, accepted 4/22/2008) Vol. 50 (2009), No. 1, pp. 77-92.

27 A quick way to Galois and strongly pure rings,
with Steve Seif and Hamid Kulosman,
Pan-American Mathematical Journal (submitted, Dec. 31, 2007, accepted 4/12/2008) Vol. 18 (2008), No. 4, pp. 39-44.

26. Skew Hadamard designs and their codes,
with Patrick Sole
Designs, Codes, and Cryptography. presented at WCC 2007, (submitted 5/30/07, accepted 9/28/07), Special issue: Coding and Cryptography. In Memory of Hans Dobbertin, WCC 2007, Vol. 49 (2008), pp. 135-145.

25. New MDS or near-MDS self-dual codes,
with T. Aaron Gulliver and Yoonjin Lee
IEEE Transactions on Inform Theory (submitted, Jan. 2007, accepted 6/12/2008), Vol. 54 (2008), No. 9, pp. 4354-4360.

24. On self-dual codes over F_5,
with Sunghyu Han,
Designs, Codes, and Cryptography (submitted 11/6/07, accepted 2/5/08), Vol. 48 (2008) No. 1., pp. 43-58.

23. Nonbinary Quantum Error-Correcting Codes from Algebraic Curves,
 with Judy L. Walker,
Discrete Math as a special issue of Com2MaC conference, July 2004, Pusan, Korea (accepted 8/9/07) Vol. 308, No. 14. pp. 3115-3124.

22. Upper bounds for the lengths of s-extremal codes over F_2, F_4, and F_2+uF_2,
with Sunghyu Han
IEEE Trans. Inform.Theory, (submitted 3/5/07, accepted 9/25/07), Vol. 54 (2008), No. 1, pp. 418-422.

21. Construction of MDS self-dual codes over Galois rings,
with Yoonjin Lee
Designs, Codes, and Cryptography (submitted 5/21/07, accepted 7/20/07), Vol. 45 (2007), No. 2, pp. 247-258.

20. A Note on Formally Self-Dual Even Codes of Length Divisible by 8
 with Vera Pless,
 Finite Fields and Their Applications, Vol. 13, No. 2, (2007), pp. 224-229.

19. Small weight codewords in LDPC codes defined by (dual) classical generalized quadrangles,
with Keith E. Mellinger and Leo Storme
Designs, Codes and Cryptography, 42 (2007), 73-92.

18. Double circulant codes from two class association schemes with Steven T. Dougherty and Patrick Sole
Advances in Mathematics of Communications, 1 (2007), 45-64.

17. s-extremal additive F_4 codes
with E. P. Bautista, Philippe Gaborit, and Judy Walker
Advances in Mathematics of Communications (full version of ISIT 2006 version), 1 (2007), 111-130

16. Explicit construction of families of LDPC codes with no 4-cycles, (ps)
with U.N. Peled, I. Perepelitsa, V. Pless and S. Friedland,
 IEEE Trans. Inform. Theory, Vol. 50 (Oct 2004), pp. 2378-2388.

15. Circulant based extremal additive self-dual codes over GF(4), (ps)
 with T. Aaron Gulliver,
 IEEE Trans. on Inform. Theory, Vol. 40 (Feb. 2004), pp. 359-366 .

14.   Euclidean and Hermitian self-dual MDS codes over large finite fields, (ps)
 with Yoonjin Lee,
 J. Combinatorial Theory, Ser. A, 105 (2004) pp. 79-95.

13.  Designs in Additive Codes over GF(4) (revised version of 38th Allerton conference)
 with Vera Pless,
 Designs, Codes and Cryptography, Vol 30, (2003), pp. 187-199.

12. Projections of binary linear codes onto larger fields,(pdf) (revised as of June 12, 2003)
 with K. Mellinger and V. Pless,
 SIAM journal on Discrete Math, Vol 16, No. 4, (2003), pp. 591-603

11. Decoding Binary R(2,5) by Hand(ps), (revised June 22, 2001)
 with Philippe Gaborit and Vera Pless,
Discrete Math. Vol. 264 (2003), pp. 55-73.

10. Construction of Some Extremal Self-Dual Codes,
with T. Aaron Gulliver and Masaaki Harada,
Discrete Math, Vol. 263 (2003), pp. 81-91.

9. New Self-Dual Codes over GF(4) with the Highest Known Minimum Weights, (full version)
IEEE Trans. Inform. Theory,  Vol. 47 (May 2001), pp. 1575-1580.

8. New extremal self-dual codes of lengths 36,38, and 58(ps), revised,
 IEEE Trans. on Inform. Theory, Vol. 47 (Jan. 2001), pp. 386-393.

7. Relation Between Weight Distribution and Combinatorial Identities(ps),
  Bulletin of the Institute of Combinatorics and its Application, Canada, Vol 31, Jan. 2001,
69-79.

Book Chapters

6. A prize problem in coding theory,
the Book for proceedings D1: Groebner, Coding, and Cryptography (submitted, April 10, 2007, accepted 6/2008), Part II, Section 5.

5. Quantum error-correcting codes from algebraic curves,
with Gretchen L. Matthews
submitted, Mar. 4, 2008, survey paper, Advances in algebraic geometry codes, ed. E. Martinez-Moro,

C. Munuera, and D. Ruano, Series on Coding Theory and Cryptology, 5, World Scientific Publishing Co.

Pte. Ltd., Hackensack, NJ, 2008

4. Remarks on s-extremal codes,
Advances in Coding Theory and Cryptology,   Series on Coding Theory and Cryptology, 2. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007 (accepted in March 2007) pp. 101-113.

Proceeding Papers (referred)

3. New quantum-error-correcting codes from Hermitian self-orthogonal codes over GF(4) (pdf), (referred)
Proceedings of the sixth international conference on Finite fiedls and applications, at Oaxaca, Mexico, May 21-25, 01. 2001, Springer Verlag (2002), pp. 209-213.

2. On Additive GF(4) Codes(ps),(referred)
with Philippe Gaborit, W. Cary Huffman, and Vera Pless,
DIMACS Workshop on Codes and Association Schemes, DIMACS Series in Discrete Math. and Theoretical Computer Science, American Mathematical Society, Vol. 56 (2001), pp. 135-149.

1. Decoding Some Doubly-Even Self-Dual [32,16,8] Codes by Hand(ps),(referred)
with  Vera Pless,
Proceedings of XXVth Ohio State-Denison conference on Codes and Designs(May, 2000), Sept. 25, 00. pp. 165-178.

Proceeding Papers (non refereed)

4. Dual cyclic codes with two zeros,
Proceedings of the 40th Allerton Conference on Communication, Control, and Computing (10/2-10/4) at Allerton, IL., Oct. 2002, pp. 1017-1023.

3. Explicit construction of LDPC codes with girth at least six,
with Uri N. Peled , Irina Perepelitsa, and Vera Pless,
Proceedings of the 40th Allerton Conference on Communication, Control, and Computing (10/2-10/4) at Allerton, IL., Oct. 2002, pp. 1024-1031.

2. Designs in Additive Codes over GF(4),
with Vera Pless,
Proceedings of the 38th Allerton Conference on Communication, Control and Computing, UIUC, Oct. 2000, pp. 1010-1018.

1. On the Classification of Extremal Additive  Codes over GF(4)(ps),
 with Philippe  Gaborit, W. Cary Huffman, and Vera Pless,
Proceedings of the 37th Allerton Conference on Communication, Control and Computing, UIUC, Sep. 1999, pp. 535-544.

 Preprint

2. A Combinatorial Construction of Low-Density Parity-Check Codes,
 with Leo Storme, preprint

1. Codes constructed from Non-Symmetric Association Schemes(ps),
preprint, Dec. 1997,


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