Construction X and its variants are known from the theory of classical error control codes. We present instances of these constructions that produce binary stabilizer quantum error control codes from arbitrary quaternary linear codes. Our construction does not require the classical linear code C that is used as the ingredient to satisfy the dual containment condition, or, equivalently, C^\perp_h is not required to satisfy the self-orthogonality condition. We prove lower bounds on the minimum distance of quantum codes obtained from our construction. We give examples of record breaking quantum codes produced from our construction. In these examples, the ingredient code C is nearly dual containing, or, equivalently, C^\perp_h is nearly self-orthogonal, by which we mean that dim(C^\perp_h)-dim(C^\perp_h \cap C) is positive but small.

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