Minc H.'s (0,1)-matrices with minimal permanents PDF By Minc H.

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Extra info for (0,1)-matrices with minimal permanents

Example text

0). The righthand term is the polynomial corresponding to the 2’s complement of the bit-pattern dm , dm−1 , . . , d , obtained by complementing each bit, followed by the addition of a unit in the least significant position. As demonstrated, the standard digit set {0, 1, . . , β − 1} can be used for a nonredundant radix-complement representation by adding the digit −1, but may only be used in the most-significant position. It is similarly possible to obtain redundant radix-complement representations based on the extended digit set {0, 1, .

Note that this is based on possibly “left-shifting” b until it is integral, and afterward shifting it back again. Thus the operator depends on the radix β, which we will assume implicitly known from the context. For the most commonly used case of β = 2 we obtain the classical 2’s complement representation. 5 (2’s complement representation) With C2c = {−1, 0, 1} and F 2cm [2, C2c ] = P ∈ P[2, C2c ] P = m+1 i= di i , di ∈ {0, 1} for ≤ i ≤ m and dm+1 = −dm , then for P ∈ F 2cm [2, C2c ] the value of the polynomial may be obtained as   m−1 d 2i if dm = 0 i i= P =  m−1 di 2i − 2m if dm = 1 i= 40 Radix polynomial representation or alternatively m−1 P = di 2i − dm 2m .

6 (2’s complement carry-save polynomials) The finite precision 2’s complement carry-save polynomials are the set F csm [2, {−2, −1, 0, 1, 2}] = P ∈ P[2, {−2, −1, 0, 1, 2}] P = m+1 i= di i , di ≥ 0 for ≤ i ≤ m and dm+1 = −dm . 8 Finite-precision and complement representations 41 for which the the value of P ∈ F csm [2, {−2, −1, 0, 1, 2}] may be determined by P = m−1 i= di 2i − dm 2m , with range −2m+1 ≤ P ≤ 2(2m − 2 ). In a string representation dm+1 need not be present, and only the digits {0, 1, 2} are needed.