Abstract Toric codes are a type of evaluation code introduced by J.P. Hansen in 2000. They are produced by evaluating (a vector space composed of) polynomials at the points of $$(\mathbb {F}_q^*)^s$$ ( F q ∗ ) s , the monomials of these polynomials being related to a certain polytope. Toric codes related to hypersimplices are the result of the evaluation of a vector space of homogeneous monomially square-free polynomials of degree d . The dimension and minimum distance of toric codes related to hypersimplices have been determined by Jaramillo et al. in (Des Codes Cryptogr 89:269–300, 2021). The next-to-minimal weight in the case $$d = 1$$ d = 1 has been determined by Jaramillo-Velez et al. in (São Paulo J Math Sci 17:188– 207, 2023), and has been determined in the cases where $$3 \le d \le \frac{s - 2}{2}$$ 3 ≤ d ≤ s - 2 2 or $$\frac{s + 2}{2} \le d < s$$ s + 2 2 ≤ d < s , by Carvalho and Patanker in 2024. In this work we characterize and determine the number of minimal (respectively, next-to-minimal) weight codewords when $$3 \le d < s$$ 3 ≤ d < s (respectively, $$3 \le d \le \frac{s - 2}{2}$$ 3 ≤ d ≤ s - 2 2 or $$\frac{s + 2}{2} \le d < s$$ s + 2 2 ≤ d < s ).

In classical coding theory, error-correcting codes are designed to protect against errors occurring at individual symbol positions in a codeword. However, in practical storage and communication systems, errors often affect multiple adjacent symbols rather than single symbols independently. To address this, symbol-pair read channels were introduced [1], and later generalized tob-symbol read channels [2] to better model such error patterns.b-Symbol read channels generalize symbol-pair read channels to account for clustered errors in modern storage and communication systems. By developing bounds and efficient codes, researchers improve data reliability in applications such as storage devices, wireless networks, and DNA-based storage. Given integersq, n, d,andb≥ 2, letAb(n,d,q) denote the largest possible code size for which there exists aq-ary code of length n with a minimumb-symbol distanced. In [3], various upper and lower bounds onAb(n,d,q) are given forb= 2. In this paper, we generalize some of these bounds to theb-symbol read channels forb> 2 and present several new bounds onAb(n,d,q). In particular, we establish the linear programming bound, a recurrence relation onAb(n,d,q), the Johnson bound (even), the restricted Johnson bound, the Gilbert-Varshamov-type bound, and the Elias bound for the metric of symbolsb,b≥ 2. Furthermore, we provide examples showing that the Gilbert–Varshamov bound established in this paper yields a stronger lower bound than the one given in [4]. Additionally, we introduce an alternative approach to derive the sphere-packing and Plotkin bounds.