By Mochizuki S.

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0123 0123 ⎢1 2 3 0⎥ ⎢1 0 3 2⎥ ⎢ ⎥ ⎥ Z4 = ⎢ ⎣2 3 0 1⎦ E4 = ⎣2 3 0 1⎦ 3012 3210 The two inequivalent 5 × 5 Latin squares (the addition table for the cyclic group Z5 and the loop L5 ). ⎡ ⎡ ⎤ ⎤ 01234 01234 ⎢1 2 3 4 0⎥ ⎢1 3 0 4 2⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ Z5 = ⎢2 3 4 0 1⎥ L5 = ⎢ ⎢2 4 3 1 0⎥ ⎣3 4 0 1 2⎦ ⎣3 0 4 2 1⎦ 40123 42103 References 1. F. Gray. Pulse code communication, March 17, 1953. S. patent no. 2,632,058. 2. C. Huﬀman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge: Cambridge University Press, 2003.

Let w(x) = j + 1. , the vectors y obtained from x by changing one nonzero component to zero. By induction, f (y) = f0 (y) for each of these neighbors, so f (x) = f0 (y) = f0 (x) − xi . This determines f (x) except when x is a two-valued vector, say x = (1111000). But then x belongs to a clique {(0111000), (1111000), (2111000)} in which every vector except x itself is already colored, either because it has weight j or because it has weight j + 1 and is three-valued. 5 GPC’s and Latin Squares A Latin square of order q is a q × q matrix with entries from X, such that every element of X appears exactly once in each row and column [3].

Corollary 2. Let n be any positive integer and k any non-negative integer such that k ≤ n. Let f1 , . , f7 be k-th order correlation immune (resp. k-resilient) functions. If two among the functions σ1 = f1 ⊕ . . ⊕ f7 , σ2 = f1 f2 ⊕ f1 f3 ⊕ . ⊕ l f6 f7 and σ4 = fij is k-th order correlation immune (resp.

### A Theory of Ordinary p-Adic Curves by Mochizuki S.

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