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Note presents a 193-self-orthogonal Latin square of order 14, completing the spectrum of r-self-orthogonal squares.
A Latin square of order n is r-self-orthogonal if it and its transpose are r-orthogonal, i.e. their superposition yields exactly r distinct ordered pairs. The spectrum of r-self-orthogonal Latin squares was determined for every order except n = 14, where the single value r = 14² − 3 = 193 remained undecided. This note exhibits a 193-self-orthogonal Latin square of order 14, settling this case and completing the spectrum of r-self-orthogonal Latin squares. A self-contained verification script is included.
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Lee et al. (2026) studied this question.
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