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(Curriculum-vitae, Partially balanced designs)

A (n, M, d; q) code is a q-ary code of M codewords, each of length n and having

distance d .Semakov and Zinoviev (1968) (op.cit.Tonchev,1989) and Sinha(1994),

Sinha& Mitra (1999) constructed q-ary codes from resolvable BIB designs and nested BIB designs. For definition of nested BIB designs see Preece (1967), Sinha and Mitra (1994).

It is known that (see, Tonchev 1989) for any equidistant (n, M, d; q) code,

d£nM(q-1)/{(M-1)q} (d=d_opt, say) where the equality is achieved if and only if M is a multiple of q and each of the symbols 0,1,…,q-1 occurs exactly M/q times in each of the Mxn matrix formed by the codewords. An equidistant code that achieves the above inequality is said to be optimal. A necessary condition for the existence of an optimal equidistant code is that d_opt be an integer, i.e., the equidistant code is not optimal, then the code with d = [d_opt ] is called nearly optimal, which is obviously the best possible code (see Miao and Kageyama,1999).

No.	n	M	d	q	nested BIBD(Preece, 1967)
1*	5	6	4	3	1
2	5	8	3	3	1
3	7	7	5	3	2
4*	7	8	6	4	2
5	7	10	4	3	2
6	7	11	5	4	2
7	11	12	10	6	8
8	11	17	9	6	8
9	13	13	10	4	10(i)
10	13	13	11	5	10(i)
11*	13	14	12	7	10(i)
12	14	8	10	3	3 (i)
13	9	9	6	3	4 (i)
14	9	9	8	5	4 (i)
15	10	10	8	4	5 (i)
16	15	10	12	4	5(ii)
17	16	16	12	4	13
18	16	16	14	6	13

*The codes marked by * are optimal and are obtained by adding an extra codeword (0,0,0...,0).

References

Miao, Y., Kageyama, S. (1999) Two classes of q-ary codes based on group divisible

association schemes, Discrete Mathematics,195,269-276.

Preece, D.A. (1967) Nested balanced incomplete block designs,

Biometrika 54,479-486.

Semakov, N.V., Zinoviev, A.A. (1968) Equidistant q-ary codes with maximal distance

and resolvable balanced incomplete block designs,

Problemi Predeatchi Informatsii 4,3-10.

Sinha, K. (1994) A class of q-ary codes, Discrete Mathematics, 126,439-940.

Sinha, K., Mitra, R.K. (1999) Construction of nested balanced block designs, rectangular

designs and q-ary codes,Annals of Combinatorics,3,71-80.

Tonchev, V.D. (1989) Combinatorial configurations, Codes, Graphs,

John Wiley & sons, New York.

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