Name for the weight function defined as the integer sum of coordinate entries from ${\mathbf F}_p$

Question

In ${\mathbb F}_p^n$, $p$ prime one may define a weight function on vectors in various ways such as Hamming, or Lee weight. (These two weights correspond nicely to the respective distances from $\bar 0$.)

If ${\mathbb F}_p =\{ 0,1, \ldots,p-1\}$ then one could alternatively (perhaps naively) define a weight function $\omega$ as the integer sum of coordinate entries: $$\omega(\bar{x}) =x_1+x_2+\ldots +x_n \;\;\;\;\;\;\;\text{(integer addition)}. $$ I suspect that this "checksum" type weight must have a name but cannot seem to find a reference.

Literature references would be most appreciated.

Answer 1

I am very familiar with the coding theory and cryptography literature and have read many papers over the years. I have never seen this referred to as anything more specific than the sum of the vector coordinates.

Sometimes it comes up more generally as $\langle a,x\rangle$ for $a=(1,\ldots,1)$ and in certain applications it is more convenient to think of this as $$ \textrm{tr}(ax), \quad a,x\in \mathbb{F}_{p^n}, $$ where the coordinates of $a$ and $x$ can be represented by a self-dual basis which sets up a bijection between $\mathbb{F}_{2^n}$ and $\mathbb{F}_p^n.$

It can of course informally be called the full parity check with respect to the all 1 vector.

Answer 2

One description is that you are using a particular lifting map $\phi:\mathbb F_p\hookrightarrow\mathbb Z$ that inverts the reduction mod $p$ map, and then your quantity is simply the $L^1$ norm on $\mathbb Z$, i.e., $$\omega(\bar x)=\bigl\|\phi(\bar x)\bigr\|_1.$$ In cryptographic constructions, this is more commonly used with a center-lifting map, i.e., choose representatives centered around $0$, $$\phi : \mathbb F_p \hookrightarrow \left\{ -\frac{p-1}{2},\ldots,\frac{p-1}{2} \right\}. $$ Of course, it's then essential to take absolute values before summing. And I'll mention that although the $L^1$ norm does come up occasionally, I think it's more common to see the $L^2$ and $L^\infty$ norms, i.e., $$ \sqrt{x_1^2+\cdots+x_n^2} \quad\text{and}\quad \max\bigl\{|x_1|,\ldots,,|x_n|\bigr\}. $$