ISSN 0032-9460, Problems of Information Transmission, 2011, Vol. 47, No. 4, pp. 327–341.
Pleiades Publishing, Inc., 2011.
Original Russian Text
A.A. Frolov, V.V. Zyablov, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 4, pp. 27–42.
Bounds on the Minimum Code Distance for
Nonbinary Codes Based on Bipartite Graphs
A. A. Frolov and V. V. Zyablov
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow
Received March 28, 2011; in ﬁnal form, September 19, 2011
Abstract—The minimum distance of codes on bipartite graphs (BG codes) over GF (q) is
studied. A new upper bound on the minimum distance of BG codes is derived. The bound
is shown to lie below the Gilbert–Varshamov bound when q ≥ 32. Since the codes based on
bipartite expander graphs (BEG codes) are a special case of BG codes and the resulting bound
is valid for any BG code, it is also valid for BEG codes. Thus, nonbinary (q ≥ 32)BGcodes
are worse than the best known linear codes. This is the key result of the work. We also obtain
a lower bound on the minimum distance of BG codes with a Reed–Solomon constituent code
and a lower bound on the minimum distance of low-density parity-check (LDPC) codes with a
Reed–Solomon constituent code. The bound for LDPC codes is very close to the Gilbert–Var-
shamov bound and lies above the upper bound for BG codes.
In this paper we consider nonbinary codes based on bipartite graphs. The idea of codes on
graphs was proposed by Tanner in . Later, expander graphs were used by Sipser and Spielman
in  to obtain asymptotically good codes with simple decoding (by “asymptotically good codes”
we mean codes whose rate and relative minimum distance are both bounded away from zero). They
suggested the term expander codes for the codes. Along with random expander graphs, explicit
constructions of graphs with a good expansion coeﬃcient [3, 4] were used in , which are called
Ramanujan graphs. In [5, 6], a special case of the Sipser–Spielman construction was considered in
which the underlying graph is bipartite. The construction is a special case of BG codes. In this
paper, distance properties of BG codes are studied.
Lower bounds on the minimum distance of binary BG codes were obtained in [7, 8]. In addition,
one can use results of [9,10], where lower bounds for generalized binary LDPC codes were obtained.
The results can easily be generalized to the case of nonbinary codes. Unfortunately, we could not
ﬁnd any work where an upper bound on the minimum distance of BG codes is obtained. In 
upper bounds on the minimum distance of binary LDPC codeweare derived. However, the authors
showed that the constructed upper bound lies below the Gilbert–Varshamov bound only at very
high rates (R>0.975), and for the remainder of the interval the bound only improves the general
upper bounds valid for all linear codes. To construct an upper bound, we use one of the methods
In the present paper, we obtain upper and lower bounds on the minimum distance of BG codes
over GF (q). Itisshownthatforanyq ≥ 32 there is an interval, expanding as q grows (not at
high rates), in which the obtained upper bound lies below the Gilbert–Varshamov bound, and thus
a fundamental result is proved: nonbinary BG codes are worse than the best known linear codes.