Boguslavsky–Tsfasman conjecture for projective point counts over finite fields

Let $S(m)=\mathbb{F}_q[x_0,\dots,x_m]$, and let $S_d(m)$ be its degree-$d$ graded component. For $m,d\geq 1$ and $1\leq r\leq \binom{m+d}{d}$, let $e_r(d,m)$ be the maximum number of $\mathbb{F}_q$-rational points of a projective algebraic set defined by $r$ linearly independent homogeneous polynomials of degree $d$ in $m+1$ variables. Let $\Omega(d,m)$ be the set of $(m+1)$-tuples of nonnegative integers summing to $d$, and let $w_r(d,m)=(\beta_1,\dots,\beta_{m+1})$ be the tuple specified in the conjecture, with $l=\min\{i\mid \beta_i\neq 0\}$. Write $\pi_j=1+q+\cdots+q^j$ for $j\geq 0$, and set $\pi_{-1}=0$. Boguslavsky–Tsfasman conjecture. If $q\geq d+1$, then

$$e_r(d,m)=\sum_{i=l}^{m}\beta_i\bigl(\pi_{m-i}-\pi_{m-i-l}\bigr)+\pi_{m-2l}.$$

The conjecture proposes an exact projective analogue of the known affine formula of Heijnen and Pellikaan for generalized Reed–Muller codes. Its status is not resolved by the supplied text.