Dong Yeol Oh

A classification of posets admitting the MacWilliams identity

In this paper, all poset structures that admit the MacWilliams identity are classified, and the MacWilliams identities for poset weight enumerators corresponding to such posets are derived. It is proved that being a hierarchical poset is a necessary and sufficient condition for a poset to admit the MacWilliams identity. An explicit relation is also derived between the P-weight distribution of a hierarchical poset code and the P~-weight distribution of the dual code.

On the nonexistence of triple-error-correcting perfect binary linear codes with a crown poset structure

Ahn et al. [Discrete Math. 268 (2003) 21-30] characterized completely the parameters of single- and error-correcting perfect linear codes with a crown poset structure by solving Ramanujan-Nagell-type Diophantine equation. In this paper, we give a shorter proof for the same result by analyzing a generator matrix of a perfect linear code. Furthermore, we combine our method with the Johnson bound in coding theory to prove that there are no triple-error-correcting perfect binary codes with a crown poset structure. tructure.

Optimal Single Deletion Correcting Code of Length Four Over an Alphabet of Even Size

We improve Levenshteins upper bound for the cardinality of a code of length four that is capable of correcting single deletions over an alphabet of even size. We also illustrate that the new upper bound is sharp. Furthermore we construct an optimal perfect code that is capable of correcting single deletions for the same parameters.

Corrigendum: Corrigendum to A classification of the structures of some Sperner families and superimposed codes

In the above mentioned article there are two errors, Theorem 4.4 and Corollary 4.5, in the paper [1]. A correct version of Theorem 4.4 may be stated as follows: Theorem 4.4. There are exactly four non-equivalent (1, 2) superimposed codes of size 9× 10. In the proof of Theorem 4.4, six superimposed codes C were given. Among the first four superimposed codes in the line 4 of page 1730, three codes, except the first one, are equivalent: the second and the third are equivalent to the fourth by permuting the columns of C, (3 4)(5 6)(8 9) and (2 3 4) (6 5 4) (8 10), respectively. Hence a correct version of Corollary 4.5 is stated as follows: Corollary 4.5. There are exactly four non-equivalent (9, 10, 4) codes with constant weight 3. I apologize to the readers for my careless errors. I would like to thank Prof. Stoyan Kapralov for pointing out these errors. Reference

Corrigendum to “A classification of the structures of some Sperner families and superimposed codes”: [Discrete Math. 306 (2006) 1722–1731]

In the above mentioned article there are two errors, Theorem 4.4 and Corollary 4.5, in the paper [1]. A correct version of Theorem 4.4 may be stated as follows: Theorem 4.4. There are exactly four non-equivalent (1, 2) superimposed codes of size 9× 10. In the proof of Theorem 4.4, six superimposed codes C were given. Among the first four superimposed codes in the line 4 of page 1730, three codes, except the first one, are equivalent: the second and the third are equivalent to the fourth by permuting the columns of C, (3 4)(5 6)(8 9) and (2 3 4) (6 5 4) (8 10), respectively. Hence a correct version of Corollary 4.5 is stated as follows: Corollary 4.5. There are exactly four non-equivalent (9, 10, 4) codes with constant weight 3. I apologize to the readers for my careless errors. I would like to thank Prof. Stoyan Kapralov for pointing out these errors. Reference