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The family of codeword stabilized codes encompasses the stabilizer codes aswell as many of the best known nonadditive codes. However, constructing optimal$n$-qubit codeword stabilized codes is made difficult by two main factors. Thefirst of these is the exponential growth with $n$ of the number of graphs onwhich a code can be based. The second is the NP-hardness of the maximum cliquesearch required to construct a code from a given graph. We address the secondof these issues through the use of a heuristic clique finding algorithm. Thisapproach has allowed us to find $((9,97\leq K\leq100,2))$ and $((11,387\leqK\leq416,2))$ codes, which are larger than any previously known codes. Toaddress the exponential growth of the search space, we demonstrate that graphsthat give large codes typically yield clique graphs with a large number ofnodes. The number of such nodes can be determined relatively efficiently, andwe demonstrate that $n$-node graphs yielding large clique graphs can be foundusing a genetic algorithm. This algorithm uses a novel spectral bisection basedcrossover operation that we demonstrate to be superior to more standardcrossover operations. Using this genetic algorithm approach, we have found$((13,18,4))$ and $((13,20,4))$ codes that are larger than any previously knowncode. We also consider codes for the amplitude damping channel. We demonstratethat for $n\leq9$, optimal codeword stabilized codes correcting a singleamplitude damping error can be found by considering standard form codes thatdetect one of only three of the $3^{n}$ possible equivalent error sets. Bycombining this error set selection with the genetic algorithm approach, we havefound $((11,68))$ and $((11,80))$ codes capable of correcting a singleamplitude damping error and $((11,4))$, $((12,4))$, $((13,8))$, and $((14,16))$codes capable of correcting two amplitude damping

Heuristic construction of codeword stabilized codes