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Source. Published pp. 226–227, Claim 3.7 and its proof. (canonical PDF).

Use the construction of lemma_3_5 and fix distinct rows $i,h$, so $r\ge2$. For $1\le j,j'\le k$,

$$|A^{(i)}_j\cap A^{(h)}_{j'}| =bq^{r-2}+l\,\mathbf1_{\{u^{(i)}_j=u^{(h)}_{j'}\}},$$

and for $1\le j\le k$,

$$|A^{(i)}_j\cap A^{(h)}_0| =bq^{r-2}+l\,\mathbf1_{\{u^{(i)}_j\notin K^{(h)}\}}.$$

These are the printed formulas (13) and (14) (p. 226). The reverse mixed-zero formula follows by exchanging the rows. Also, as an addition not in the printed claim,

$$|A^{(i)}_0\cap A^{(h)}_0| =bq^{r-2}+l\bigl(s-|K^{(i)}\cup K^{(h)}|\bigr).$$

Proof.

Fixing two row labels leaves exactly $q^{r-2}$ choices for the other entries of a word $w$. Therefore the added blocks contribute $bq^{r-2}$ to every one of these intersections.

For positive labels, the two core parts are single blocks $L_{u^{(i)}_j}$ and $L_{u^{(h)}_{j'}}$, whose intersection has size $l$ if the indices agree and zero otherwise. In a mixed-zero intersection, $L_{u^{(i)}_j}$ lies in the other row's zero part precisely when its index is absent from $K^{(h)}$. For two zero labels, the common core blocks are exactly those indexed outside $K^{(i)}\cup K^{(h)}$. The core and added blocks are disjoint, so adding these contributions proves all formulas.

Source precision.

The mixed-zero displayed unions on p. 227 contain inconsistent dummy-label conditions. The formulas above impose the actual labels of the two parts. The two-zero formula is an elementary completion of the same count; its coefficient in the squared-distance sum is zero because $a_0=0$.

Bears on. #174.