An Introduction to Abstract Algebra via Applications by David R. Finston and Patrick J. Morandi

By David R. Finston and Patrick J. Morandi

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The addition operation is an ordinary binary operation on V . Scalar multiplication is somewhat di¤erent. This operation takes a pair consisting of a scalar and a vector and returns a vector. If F is the set of scalars, then this operation is really a function from F V to V . 1. Let F be a …eld. An F -vector space is a nonempty set V together with an operation + on V and an operation of scalar multiplication, such that for all u; v; w 2 V and ; 2 F, u + v = v + u; 59 60 CHAPTER 4. LINEAR ALGEBRA AND LINEAR CODES u + (v + w) = (u + v) + w; there is a vector 0 with u + 0 = u for any u 2 V ; for every u 2 V there is a vector u 2 V with u + ( u) = 0; (u + v) = u + v; ( + )u = u + u; ( )u = ( u) 1 u = u.

Show that s = u + bi . Conclude that wt(s) > 3 and wt(s + bi ) 2, and that v = w + (s + bi ; ei ). 3. Suppose that e = (0; u) with wt(u) 3. Show that s is the sum of at most three of the bi and that u = Bs. Conclude that wt(s) > 3 but wt(Bs) 3, and that v = w + (0; Bs). 4. Suppose that e = (ei ; u) with wt(u) 2. Show that s = ei + Bu, and that Bs = bi + u. Conclude that wt(s) > 3, wt(s + bi ) > 2 for any i, and that e = (ei ; Bs + bi ), so v = w + (ei ; Bs + bi ). These four problems show how, given any possibility of an error vector e having weight at most 3, how we can determine it in terms of the syndrome s.

We denote the i-th row (and column) of B by bi , and ei denotes the 12-tuple whose i-th component is 1 and whose other components are 0. Let v be the closest codeword to w, write v = w + e, and suppose that wt(e) 3. Recall that B 2 = I and B T = B. Furthermore, a tedious check of the rows of B show that (i) wt(bi ) 7 for all i; (ii) wt(bi + bj ) 6 if i 6= j; (iii) wt(bi + bj + bk ) 5 for all i; j; k. Since B T = B, the i-th column of B is bi , and so Bei = bi . You are free to use these facts. 1.

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