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I am reading the Hamming Quasi-Cyclic (HQC) specification and just want to clarify a notation they are using.

In the paragraph before Definition 2.1.14 (2-QCSD-P Distribution), for $b_1 \in \{0,1\} $ they define a set

$$\mathbb{F}^{n}_{2,b1}= \{h \in \mathbb{F}_2^n \, \, \text{s.t.} \, \, h(1) = b_1 \mod 2 \} $$ i.e. binary vectors of length n and parity $b_1$.

The notation $h(1)$ seems to me as a way of "evaluating" $h$ the first coordinate but the text after clarifies:

i.e. binary vectors of length n and parity $b_1$.

So in this case the notation $h(1)$ is referring to the $\ell_1$ norm of $h$, and it being equal to $b_1 \mod 2$.

Haven't seen this notation as an evaluation and wanted to clarify if it is common.

grover
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Per the preliminaries in section 2.1

Let $\mathcal V$ denotes a vector space of dimension $n$ over $\mathbb F_2$ for some positive $n\in \mathbb Z$. Elements of $\mathcal V$ can be interchangeably considered as row vectors or polynomials in $\mathcal R =\mathbb F_2[X]/(X^n −1)$. Vectors/Polynomials (resp. matrices) will be represented by lower-case (resp. upper-case) bold letters

In particular then, $\mathbf h(1)$ should be interpreted as evaluating the polynomial in $\mathcal R$ whose coefficients are those of the vector $\mathbf h$ at 1. Note then that evaluating the polynomial $\mathbf h(X)$ at $X=1$ gives the parity of the Hamming weight of the vector $\mathbf h$.

Daniel S
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