Let ${\displaystyle f_{1},\dots ,f_{2^{n}}:\{0,1\}^{k}\to \{0,1\}}$  for ${\displaystyle k=\log n}$  be the list of all functions from ${\displaystyle \{0,1\}^{k}\to \{0,1\}}$ . Then the long code encoding of a message ${\displaystyle x\in \{0,1\}^{k}}$  is the string ${\displaystyle f_{1}(x)\circ f_{2}(x)\circ \dots \circ f_{2^{n}}(x)}$  where ${\displaystyle \circ }$  denotes concatenation of strings. This string has length ${\displaystyle 2^{n}=2^{2^{k}}}$ .

The Walsh-Hadamard code is a subcode of the long code, and can be obtained by only using functions ${\displaystyle f_{i}}$  that are linear functions when interpreted as functions ${\displaystyle \mathbb {F} _{2}^{k}\to \mathbb {F} _{2}}$  on the finite field with two elements. Since there are only ${\displaystyle 2^{k}}$  such functions, the block length of the Walsh-Hadamard code is ${\displaystyle 2^{k}}$ .

An equivalent definition of the long code is as follows: The Long code encoding of ${\displaystyle j\in [n]}$  is defined to be the truth table of the Boolean dictatorship function on the ${\displaystyle j}$ th coordinate, i.e., the truth table of ${\displaystyle f:\{0,1\}^{n}\to \{0,1\}}$  with ${\displaystyle f(x_{1},\dots ,x_{n})=x_{j}}$ .^[1] Thus, the Long code encodes a ${\displaystyle (\log n)}$ -bit string as a ${\displaystyle 2^{n}}$ -bit string.

The long code does not contain repetitions, in the sense that the function ${\displaystyle f_{i}}$  computing the ${\displaystyle i}$ th bit of the output is different from any function ${\displaystyle f_{j}}$  computing the ${\displaystyle j}$ th bit of the output for ${\displaystyle j\neq i}$ . Among all codes that do not contain repetitions, the long code has the longest possible output. Moreover, it contains all non-repeating codes as a subcode.