By Hugo Steinhaus
Problems-with instructive solutions-on numbers, equations, polygons, polyhedra, and plenty of different themes. Very demanding. extra thirteen difficulties with no strategies.
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Therefore -2P n >-XC(P) > - in , p < 1/2. (47) 1 Now we can show that when (-ln ) (2a) a- 1, C(pa ) < ,a(p) (48) For, by (47), 3C( a ) < _2pa in pa = pa . 2a(-ln ) ca(p) > a(-ln )a; but 2ax < xa a- 1 (2a) when x which proves Eq. 48. We note that when p _<1/e2 , a 4, this condition is always satis- fied. ) Finally, when x > y > 0, and a > 1, (x-y) a = Xa -( ) >xa I >xa( -) = xa _ ya. (49) We are now ready to construct our many-stage concatenated code. l. -- --111- __ -E Pr(e) Po= e (50) E > 1 We now apply to this superchannel an RS code of dimensionless rate (1-2Z) and length N 1 , achieving a probability of error, from (46), of Prl(e) e -N 1 [E -X(P) ] -E 1 e (51) Assume 3E - X(P) > 0, and define a to satisfy N 1 [pE -C(p)] Ea = E1 thus a = In N 1 In [E - (p)] E + ln E (52) We assume that 3 ~ 1/e2 (53) and 4 ~ a N 1 (1- 2z), and we shall prove the theorem only for these conditions.
Thus the smallest non-null set must be an inforrra- Since every linear code has a smallest non-null set, every linear code has an k information set and, for some k, q code words. In fact, every non-null set of size k tion set. is an information set, since to each of the q assignment of elements to those k places. code words must correspond a different We say such a code has k information sym- bols, n - k check symbols, and dimensionless rate k/n, and call it an (n,k) code on GF (q). If the minimum distance of a code is d, then the minimum weight of any non-zero code word is d, and the largest null set has size n - d.
Are members of some field. The determinant of this matrix, D, also a member of the field, is a polynomial in the a i in which no a i appears to a power greater than n. Furthermore, since the determinant is zero if any two rows are the same, this polynomial must contain as factors a i - aj, all i D = D' j, so that I (a-a). ) contains each a to the n t h power, so that D' can only i>j 2 n 'a2 a n in this polynomial must be one, be a constant. ---.. ) -. 11_1- . Y·IIIILIIPII -CII^II(-l·P*I-_··I. - I-L_--_III··I*LI_^--lsllCII - 111--------lllllslII Then ai - a.