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4 Let k, t be two positive integers with k ≥ t. Suppose there exists a set of k vectors αi = (ci1 , ci2 , · · · , cit ) ∈ Ftq , 1 ≤ i ≤ k and any t vectors in the set are linearly independent. Set S = {1, 2, · · · , k}, M = Fq , and E = Ftq . For any source state i and encoding rule e = (a1 , a2 , · · · , at ) ∈ Ftq define t e(i) = j=1 cij aj . Prove that the code constructed is t-fold perfect with P0 = P1 = · · · = Pt−1 = 1/q if the encoding rules have a uniform probability distribution. 5 Let F2q be the vector space of dimension 2 over the finite field Fq .

For a given f ∈ ER , let ET (f ) denote the random variable of valid encoding rules of f ; it takes its values from ET (f ). Similarly, for a given e ∈ ET , let ER (e) denote the random variable of decoding rules of which e is valid; it takes its values from ER (e). 2 For any given f ∈ ER , suppose that ET (f ) has a uniform probability distribution and PRr = 2H(ET |ER ,M r+1 )−H(ET |ER ,M r ) , 0 ≤ r ≤ t − 1. Then for any mr ∗ m ∈ MRr+1 , f ∈ ER (mr ∗ m), PRr = Copyright 2006 by Taylor & Francis Group, LLC |ET (f, mr ∗ m)| .

2 A orthogonal array OA(nt , k, n, t) is an r − (kn, nt , k; nt−r , 0) design for 2 ≤ r ≤ t and is a 1 − (kn, nt , k, nt−1 ) design as well. 5 Authentication Codes and Combinatorial Designs Combinatorial Bound In this section we discuss the combinatorial bound on Pr . Recall that |S | = k, |M | = v, and |E | = b. 1}. Also, we know that only when m = mi for 1 ≤ i ≤ r it is possible for P (m|mr ) = p(e|mr ) e∈E (mr ∗m) to be nonzero. 21) holds if and only if P (m|mr ) = p(e|mr ) = e∈E (mr ∗m) k−r v−r for any mr = (m1 , m2 , · · · , mr ) ∈ M r and any m ∈ M with m = mi , 1 ≤ i ≤ r.

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