On strongly walk regular graphs, triple sum sets and their codes

Abstract Strongly walk regular graphs (SWRGs or s -SWRGs) form a natural generalization of strongly (SRGs) where paths length 2 are replaced by . They can be constructed as coset the duals projective three-weight codes whose weights satisfy certain equation. We provide classifications feasible parameters these in binary and ternary case for medium size code lengths. For case, divisibility is investigated several general results shown. It known that an -SWRG has at most 4 distinct eigenvalues $$k> \theta _1> _2 > _3$$ k > θ 1 2 3 , triple $$(\theta _1, _2, _3)$$ ( , ) satisfies homogeneous polynomial equation degree $$s - 2$$ s - (Van Dam, Omidi, 2013). This defines plane algebraic curve; we use methods from algorithmic arithmetic geometry to show = 5$$ = 5 7$$ 7 there only obvious solutions, conjecture this remain true all (odd) \ge 9$$ ≥ 9