be zero(s), and vice versa withs1. Other states are abbreviated with their first initials in Equation5.11.
h
1h
2h
1h
2h
1h h
h3v
1v
2v v
1Con f i g ur ati on1 Con f i g ur ati on2
0 12 21 0 12 21
h1 h2 Vh− h v1 v2 h v1 v2 h h v h2 h v h1 h
s0 s0 0+ p 0 0 e 0 0 e s1 0 v e 0 v e
s0 s0 0+e p 0 0 s1 0 0 s1 p 0 v s1 0 v s1
s0 s0 0+e+ p 0 0 p 0 0 p p 0 v p 0 v p
s0 s0 0+1 s0 0 0 v 0 0 v s0 0 v v 0 v v s0 s0 0+1e s0 0 0 s0 0 0 s0 s0 0 v s0 0 v s0 s0 s0 0+1e+ s0 0 0 s0 0 0 s0 s0 0 v s0 0 v s0
s0 s1 0+ p 0 1 v 0 1 v s1 0 e v 1 e v
s0 s1 0+e p 0 1 s0 0 1 s0 p 0 e s0 1 e s0 s0 s1 0+e+ p 0 1 s0 0 1 s0 p 0 e s0 1 e s0 s0 s1 0+1 s0 0 1 e 0 1 e s0 0 e e 1 e e s0 s1 0+1e s0 0 1 e 0 1 e s0 0 e e 1 e e s0 s1 0+1e+ s0 0 1 e 0 1 e s0 0 e e 1 e e
s1 s0 0+ p 1 0 v 1 0 v s1 1 e e 0 e e
s1 s0 0+e p 1 0 s0 1 0 s0 p 1 e s1 0 e s1 s1 s0 0+e+ p 1 0 s0 1 0 s0 p 1 e p 0 e p s1 s0 0+1 s0 1 0 e 1 0 e s0 1 e v 0 e v s1 s0 0+1e s0 1 0 e 1 0 e s0 1 e s0 0 e s0 s1 s0 0+1e+ s0 1 0 e 1 0 e s0 1 e s0 0 e s0
s1 s1 0+ p 1 1 e 1 1 e s1 1 v v 1 v v
s1 s1 0+e p 1 1 e 1 1 e p 1 v s0 1 v s0 s1 s1 0+e+ p 1 1 e 1 1 e p 1 v s0 1 v s0
s1 s1 0+1 s0 1 1 e 1 1 e s0 1 v e 1 v e s1 s1 0+1e s0 1 1 e 1 1 e s0 1 v e 1 v e s1 s1 0+1e+ s0 1 1 e 1 1 e s0 1 v e 1 v e
Table 1: The first two configurations
Having proven that SOLVEreturns a unique answer regardless of the order in which the sol ve nodes are processed, we address the issue of handling the stopping sets. SOLVE
is called (recursively) by another procedure called DECODE, shown in Listing2. The recursion is initiated by executing DECODE(X,1). The parametermaxsol controls the maximum number of results returned by DECODE, making it a variablelist decoder.
Line5requires explanation: the row, column, block, and erasures mentioned there refer to theq-aryS, not the binaryS′. For instance, if the first row has only three erasures, while the other rows, columns, and blocks have more erasures, thenxrefers to the first
row. Ifq=9 and the missing numbers inxare 3, 4, and 7, then there are 3! possible ways to place these numbers intox, each potentially leading to a solution, which is stored in Y. Eachy∈Y is then passed as a context to SOLVE, which tries to find a solution within y.
In conclusion, in this paper we have presented an iterative algorithm that can be used to decode (and solve) the Latin and Sudoku codes (and puzzles). We proved that the algorithm returns consistent solutions regardless of the path taken to compute them.
An interesting question for future research is whether the same decoding algorithm can be adapted to solve Sudoku puzzles with symbol errors (instead of erasures).
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6
Additional Topics
HISchapter contains two sections that discuss additional topics in the area of (1) error-correcting codes and their relationships to combinatorial structures, and (2) application of message-passing algorithms in games of perfect information.
In Chapter 5we discussed the close relationship between binary LDPC codes and Latin (or Sudoku) Squares. In the first section, we will discuss the close relationship between MDC codes and ternary Latin hypersquares. We prove that everyq-ary MDS code of lengthnand distance 2 can be uniquely represented by a Latin square of order q. By counting the total number of the latter, we count the total number of the former.
In our analysis, we define a Latin hypersquare as a generalization of a Latin square.
Suppose we represent each letter in the ternary alphabet by three colored cubes: white, grey, and black. Hypercubes of dimensions 0, 1, 2, and 3 can be represented by a single cube, an array of three cubes, a 3×3 square of nine cubes, and a 3×3×3 block of 27 cubes, as shown in Figure6.1.
(a) (b) (c) (d)