To circumvent the size difference between $S$ and $v$, one can use [JPL codes][1]. These are a XOR (which is modulo $2$ addition) of two or more maximal length sequences of coprime sizes.

This appears to be the exact but permuted same as I advised in my comments. For two bit sequences of coprime sizes $2^n - 1$ and $2^m - 1$, make the size the period of each sequence as periodic infinite bit sequence. Adding them with XOR gives a new infinite bit sequence with the product of both periods as period (since they are coprime). Take one period of this sequence as the resulting PRBS $x$. This can be used as circulant matrix $K$, but on a permuted $v$: $v_i$ must then be the output of the ($i$ mod $2^n - 1$)th sampler on timeframe $i$ mod $2^m - 1$.

By writing the circulant matrix as a sum of powers of another cyclic permutation $P$ rather than $(1 \space 2 \space ... \space 5080)$, "the tensor XOR addition" matrix $M=(S_{ij}+T_{kl})_{(i,k)(j,l)}$ (mod 2) with index order first all samplers on the first timeframe, then the second timeframe etc. can be retrieved. This matrix retains the autocorrelation of $x$. However $40$ timeframes is a bad example, in order to use JPL codes you need an amount of timeframes that is a product of coprime factors $2^{n_i} -1$ such as $31$ or $63$ or $7×15=105$. It must also be coprime to the amount of samplers.

You can either reorder $v$ or find $P$, it is defined as the cycle with on the $i$th zero-based position the number $q+iL$ (modulo the size) where $q$ between $0$ and $L-1$ equals $i$ modulo $L$. Reordering $v$ means applying the inverse of $P$ to it, but this means also the results after deconvolution are permuted. In formulas, $M=PKP^T$ and reordered $v$ is $P^Tv$ so that $KP^Tv=P^TMv$. The autocorrelation of $K$ is the same as that of $M$ since it's merely permuted.

I'm not sure what to do if you can't exactly control the amount of timeframes, but I think it's fine to add zeroes at the end of $v$ to get the right dimension.

  [1]: https://en.m.wikipedia.org/wiki/JPL_sequence