Codes from Neighbourhood Designs of the Graphs $$GP(q,\frac{q-1}{2})$$ G P ( q , q - 1 2 ) with q Odd

Codes from neighbourhood designs of generalized Paley graphs $$GP(q,k)$$GP(q,k) of a finite field of order q and their complements are investigated. With certain q and k, we obtain the main parameters of such codes and their duals including bases of minimum-weight vectors for the codes.