What are Walsh Hadamard Codes?

Question

Which of the following bit strings are Walsh-Hadamard (WH) codewords?

00001111

11111111

01011011

01011010

Can somebody explain me what are walsh-hadamard codewords?

Answer 1

A walsh-hadamard codeword is a bitstring that is the truth table of a first degree multivariate binary polynomial, interpreted modulo 2. Also known as a first order Reed-Muller code.

So, your words have length 8. Thus 3 variables. Normally one writes them in the standard order of input variables with $x_3x_2x_1=000,001,010,011,100,101,110,110$.

So such a codeword is of the form $a_0+a_1 x_1+a_2 x_2 + a_3 x_3\pmod 2,$ with $a_i\in\{0,1\}$ and thus unless it is a constant codeword ($a_1=a_2=a_3=0,$ it will have Hamming weight (number of nonzero entries) exactly equal to 4.

Thus for example your first pattern $00001111$ is exactly $1+x_3,$ so it is such a codeword. You can work out the rest.