The main objective of this paper is to establish a new connection between the Hermitian rank-1 projector solutions of the Euclidean $\mathbb {C}P^{2S}$ sigma model in two dimensions and the particular hypergeometric orthogonal polynomials called Krawtchouk polynomials. We show that any Veronese subsequent analytical solutions of the $\mathbb {C}P^{2S}$ model, defined on the Riemann sphere and having a finite action, can be explicitly parametrized in terms of these polynomials. We apply these results to the analysis of surfaces associated with $\mathbb {C}P^{2S}$ models defined using the generalized Weierstrass formula for immersion. We show that these surfaces are homeomorphic to spheres in the $\mathfrak {su}(2s+1)$ algebra and express several other geometrical characteristics in terms of the Krawtchouk polynomials. Finally, a connection between the $\mathfrak {su}(2)$ spin-s representation and the $\mathbb {C}P^{2S}$ model is explored in detail.