By Paul P. Martin

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6. 1). 2) We are now ready to explain coset decoding. Note that a given standard array A determines, for each word y in Fqn , a coset leader eA (y) (the first word in the row of y); and a codeword cA (y) (the first word in the column of y). For example, the coset leader associated to 1010 in the array above is 0100. Thus if we receive y, we may associate two other words to it, related by eA (y) = y − cA (y) Coset decoding: if we receive y, we decode it as the codeword cA (y) appearing in the column containing y.

Thus Zp is a set with + and × which are commutative and associative, distributive... 38. Exercise. 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]? If we are working in Z5 then: Yes! [2][3] = [6] = [1]. And [4][4] = [16] = [1]. 39. Z5 is a field.

When is the code C ∪ C ′ also linear? 46. Picking a code at random from P (F n ), it is likely to be non-linear. However “most of the codes currently studied and used are linear” (Jones and Jones, 2000). We will now see why. 56 CHAPTER 3. CODING THEORY When C ⊂ F n is linear, and of dimension k as a vector space, then M = |C| = |F |k . We call C a linear [n, k]-code. 47. The rate of a code is R = R(C) = logq M n so for a linear code R = k/n Thus the bigger k is, the more information we transmit; the bigger n is, the longer it takes to transmit.