Table of Contents

## What is binary polynomial?

Finite fields of order 2m are called binary fields or characteristic-two finite fields. The elements of GF(2m) are binary polynomials, i.e. polynomials whose coefficients are either 0 or 1. There are 2m such polynomials in the field and the degree of each polynomial is no more than m-1.

**What is polynomial representation?**

The elements of F2m are polynomials of degree less than m, with coefficients in F2; that is, {am-1xm-1 + am-2xm-2 + + a2x2 + a1x + a0 | ai = 0 or 1}. Some computations involve an polynomial f(x) = xm + fm-1xm-1 + fm-2xm-2 + …

### What is the polynomial representation of 0110110?

Now that we have a generator polynomial we can use it to encode our message, [0110110]. The polynomial corresponding to this vector is x + x2 + x4 + x5. To encode we multiply this by g(x). Thus the word [0110110] is encoded to the codeword [011010111100010].

**How are binary numbers and polynomials related in math?**

Binary numbers are simply numbers expressed in Radix 2. The main relationship between numbers and polynomials is that the coefficients of a polynomial may be numeric. For example, if I say 3 x 2 + 7 x + 5 = 0, the coefficients 3, 7, and 5 are numeric. I could write 3, 7, and 5 in binary as 011 2, 111 2, and 101 2.

#### Which is the binary form of a quadratic polynomial?

In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables q ( x , y ) = a x 2 + b x y + c y 2 , {displaystyle q(x,y)=ax^{2}+bxy+cy^{2},,} where a , b , c are the coefficients .

**What are the invariants of a binary form?**

The invariants of a binary form form a graded algebra, and Gordan (1868) proved that this algebra is finitely generated if the base field is the complex numbers. Forms of degrees 2, 3, 4, 5, 6, 7, 8, 9, 10 are sometimes called quadrics, cubic, quartics, quintics, sextics, septics or septimics, octics or octavics, nonics, and decics or decimics.

## How to find linear combinations of polynomial code words?

This, as every polynomial code, is indeed a linear code, i.e., linear combinations of code words are again code words. In a case like this where the field is GF (2), linear combinations are found by taking the XOR of the codewords expressed in binary form (e.g. 00111 XOR 10010 = 10101).