Number Theory: An Introduction to Mathematics

246 V Hadamard’s Determinant Problem andr+s=m,then AtA= [ (n− 2 )Ir+ 2 Jr 0 0 (n− 2 )Is+ 2 Js ] . Thus the upper bound in Propos ...

6 Designs 247 and x^2 +y^2 =z^2 +w^2 = 2 n− 2 , xz+yw= 0. Adding, we obtainx^2 =w^2 and hencex=w. Thusz^2 =y^2 and actuallyz=−y, ...

248 V Hadamard’s Determinant Problem 1 2 3 4 5 6 7 Fig. 1.The Fano plane. Av×bmatrixA=(αij)of 0’s and 1’s is the incidence matri ...

6 Designs 249 A 2-design is said to besquareor (more commonly, but misleadingly) ‘symmetric’ ifb=v, i.e. if the number of blocks ...

250 V Hadamard’s Determinant Problem We now consider the relationship between designs and Hadamard’s determinant problem. By pas ...

7 Groups and Codes 251 λ′=λ(v−t+ 1 )/(k−t+ 1 ) does not depend on the choice ofSand at-(v,k,λ)design (P,B)isalsoa(t− 1 )- (v,k,λ ...

252 V Hadamard’s Determinant Problem Twon×nHadamard matricesH 1 ,H 2 are said to beequivalentif one may be obtained from the oth ...

7 Groups and Codes 253 Ifρ 1 =±3, then all elements of the same triple of columns in ther-th row have the same sign and orthogon ...

254 V Hadamard’s Determinant Problem the same second triple of elements as the 4-th row, and that the 11-th and 12-th rows have ...

7 Groups and Codes 255 (ii)|x+y|≤|x|+|y|. The vector spaceVacquires the structure of a metric space if we define the (Hamming) d ...

256 V Hadamard’s Determinant Problem be denoted mathematically by 0 or 1. On account of noise the message received may differ sl ...

8 Further Remarks 257 Hadamard’s original paper of 1893 is reproduced in [16]. Surveys on Hadamard matrices have been given by H ...

258 V Hadamard’s Determinant Problem (ii) If there exists av×vmatrixAof integers such that AAt=(k−λ)I+λJ,JA=kJ, then every entry ...

9 Selected References 259 [18] M. Harwit and N.J.A. Sloane,Hadamard transform optics, Academic Press, New York, 1979. [19] A. He ...

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VI Hensel’sp-adic Numbers...................................... The ringZof all integers has a very similar algebraic structure ...

262 VI Hensel’sp-adic Numbers A non-archimedean absolute value is indeed an absolute value, since(V1)implies that (V3)′is a stre ...

1 Valued Fields 263 (iv) LetF =K((t))be the field of all formal Laurent series f(t)= ∑ n∈Zαnt n with coefficientsαn∈Ksuch thatαn ...

264 VI Hensel’sp-adic Numbers Proposition 2Let F be a field with an absolute value||. Then the following properties are equivale ...

2 Equivalence 265 2 Equivalence Ifλ,μ,αare positive real numbers withα<1, then ( λ λ+μ )α + ( μ λ+μ )α > λ λ+μ + μ λ+μ = 1 ...