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In $1977$, G. Bennett proved, by means of non-deterministic methods, aninequality which plays a fundamental role in a series of optimization problems.More precisely, Bennett's inequality shows that, for $p_{1},p_{2}\in\lbrack1,\infty]$ and all positive integers $n_{1},n_{2}$, there exists abilinear form $A_{n_{1},n_{2}}\colon\left( \mathbb{R}^{n_{1}},\left\Vert\cdot\right\Vert _{p_{1}}\right) \times\left( \mathbb{R}^{n_{2}},\left\Vert\cdot\right\Vert _{p_{2}}\right) \longrightarrow\mathbb{R}$ with coefficients$\pm1$ satisfying \[ \left\Vert A_{n_{1},n_{2}}\right\Vert \leqC_{p_{1},p_{2}}\max\left\{ n_{1}^{1-\frac{1}{p_{1}}}n_{2}^{\max\left\{\frac{1}{2}-\frac{1}{p_{2} },0\right\}},n_{2}^{1-\frac{1}{p_{2}}}n_{1}^{\max\left\{ \frac{1}{2}-\frac{1}{p_{1}},0\right\} }\right\} \] for a certain constant$C_{p_{1},p_{2}}$ depending just on $p_{1},p_{2}$; moreover, the exponents of$n_{1},n_{2}$ cannot be improved. In this paper, using a constructive approach,we prove that $C_{p_{1},p_{2}}\leq\sqrt{8/5}$ whenever $p_{1},p_{2}\in\left[2,\infty\right] $ or $p_{1}=p_{2}=p\in\left[ 1,\infty\right] $. Our techniquesare applied to provide new upper bounds for the constants of a combinatorialgame, known as Gale--Berlekamp switching game or unbalancing lights problem. Asa consequence, we improve estimates obtained by Brown and Spencer in $1971$ andby Carlson and Stolarski in $2004$.

Upper bounds for the constants of Bennett's inequality and the Gale--Berlekamp switching game