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By P. P. Martin

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3) is a 2-ary (5,4,3)-code: M r 5 n 1) = 24 (q − 1)r = 4(1 + 1 r while |S 5 | = 25 = 32, so not perfect. )7 , is a repetition code. 1} Clearly this is a (2t + 1, 2, 2t + 1)-code. 1). 1). Now, why did we only include the d odd cases in our table of A2 (n, d)? For A2 (n, d) we can deduce the even d cases from the odd. 19. The weight of a string x ∈ S n is w(x) = #non-zero entries in x 7 and roadies 44 CHAPTER 3. g. w(011) = 2 = w(10010). 20. Suppose x, y ∈ S n both have even weight. Then d(x, y) is even.

But even if q, (n, M, d) passes both bounds it does not follow that such a code exists. ) Which value of t do we use in the BP bound? The largest t such that 2t + 1 ≤ d, that is, t ≤ (d − 1)/2. The largest integer not exceeding z ∈ R is written ⌊z⌋ (‘Floor function’). So use 1 t = ⌊ (d − 1)⌋ 2 So the BP theorem implies Aq (n, d) ≤ ⌊ qn ⌊(d−1)/2⌋ r=0 n r (q − 1)r ⌋ since Aq (n, d) is an integer by definition. ) 42 CHAPTER 3. CODING THEORY n 5 6 7 8 9 10 11 12 13 14 15 16 17 .. 47 .. 1: Table of known values for A2 (n, d), and some bounds.

CODING THEORY then a′ + b′ is congruent to c. p (where at this stage p is any natural number). p is denoted Zp . Thus Zp is a set with + and × which are commutative and associative, distributive... 33. Check this! with additive and multiplicative identity; and additive inverse. Example: For p = 5 the additive inverses of [0], [1], ... are given by [0] + [0] = [0] [1] + [4] = [0] [2] + [3] = [0] so that [0]=-[0]; [4]=-[1] and [3]=-[2]. What about multiplicative inverses? Is there an [x] such that [2][x] = [1]?

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