It lS found that Moyles and Thompson's algorithm contains some mistakes.An efficmnt algorLthm for finding a mlmmal eqmvalent graph (MEG) is presented The algorithm proceeds with the following steps First, all the strongly connected (s c ) components are found.Then the set of vertmes is reordered such that the set of vertices in an s c component is ordered by consecutive integers The rows and columns of the adjacency matrix are permuted accordingly Then an MEG for each s c. component is found Finally, the parallel and the superfluous edges are removed KEY WORDS AND PHRASES' digraph, algorithm, minimal equivalent graph, adjacency matrix, acychc digraph, condensed digraph CR CATLGORILS' 5 32 PrelzminariesLet G = (V, E) be a digraph, where V as the set of all the vertices where [ V I = N, and E is the set of all the edges in G, where I E I = M.For M1 vertices v, and v~ of G, if there is an elementary path from v, to v~, it is said that v, R v~.All the paths in this paper refer to elementary directed paths.An s.c.digraph is one such that for all v, and v~ E V, v, R vj and v, R re.An acyclic digraph is one such that if v, R v,, then vj,R v,.An acyclic digraph contains no cycles.The following defines an MEG of a digraph.

This correspondence presents a class of shortened cyclic codes that are suitable for application on a compound channel. The Chinese remainder theorem is applied to the proof of the error-correcting properties of this class of shortened cyclic codes.

This correspondence will show that Iwadare-type-B1recurrent codes [1] are reversible. They are capable of correcting the same set of burst error patterns in both directions--forwards and backwards. IfS(D)is the syndrome polynomial of Iwadare code sequences in the forward direction, thenD^L S^* (D)[where(L + 1)is the decoding delay andS^* (D), the reversed polynomial ofS (D)] is the syndrome polynomial of Iwadare code sequences in the backward direction. The circuit for the generation ofS(D)can be easily modified to generateD^L S^* (D). The decoders for both directions are essentially the same.

Recurrent codes for burst-error correction are defined to be a type B1 code. A type B1 code is capable of correcting all the single burst errors of lengthlor less. A type B2 code is a subclass of a type B1 code with the restriction that the single burst error of lengthlor less occurs withinrconsecutive blocks (it is assumed thatlis divisible bybandl = rb). In this paper, the author will present a class ofq-nary recurrent codes--both type B1 and B2 codes--for burst-error correction. The construction procedures for this class of codes are simple and systematic. An interesting relation exists between a type B1 and a type B2 code. If a code is type B2 (with block lengthb, burst-error-correction capabilityl = rb), then by reducing the block length by one symbol, the type B2 code becomes a type B1 code, while the burst-error-correction capability remains the same. Both types of codes can be used for the correction of binary burst errors. The correction of binary burst errors will be briefly discussed in this paper.

Many data-transmission and data-storage systems are corrupted by disturbances of both the burst type and the random type. In this paper two classes of error-correcting codes for use with these systems have been constructed analytically and with the aid of a computer. Although not optimal, these codes do have the advantage of ease of implementation,