Equally spaced polynomial
Encyclopedia
An equally spaced polynomial (ESP) is a polynomial
used in finite field
s, specifically GF(2) (binary
).
An s-ESP of degree sm can be written as:
for 
or
A 1-ESP is known as an all one polynomial
and has additional properties including the above.
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
used in finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s, specifically GF(2) (binary
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
).
An s-ESP of degree sm can be written as:
for or
Properties
Over GF(2) the ESP has many interesting properties, including:- The Hamming weightHamming weightThe Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string of bits, this is the number of 1's in the string...
of the ESP is m + 1.
A 1-ESP is known as an all one polynomial
All one polynomial
An all one polynomial is a polynomial used in finite fields, specifically GF . The AOP is a 1-equally spaced polynomial.An AOP of degree m has all terms from xm to x0 with coefficients of 1, and can be written as...
and has additional properties including the above.