Help for algebraic structures!

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Help for algebraic structures!

A linear code C of length 7 is supposed to contain two pairs of non-trivial words adding up to one; i.e., non-zero, pairwise different x, y, z, t ∈ F^7base2 such that both x + y and z + t are equal to 1 = 1111111.

Find a lower bound for k = dim C without using any error-correction arguments.

Is your bound the highest possible dimension ensuring at least one error can be corrected?