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The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. Despite the large body of literature in quantum coding theory, many important questions, especially those centering on the issue of “good codes” are unresolved. In this dissertation the dominant underlying theme is that of constructing good quantum codes. It approaches this problem from three rather different but not exclusive strategies. Broadly, its contribution to the theory of quantum error correction is threefold. Firstly, it extends the framework of an important class of quantum codes – nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. In particular it provides many explicit constructions of stabilizer codes, most notably it simplifies the criteria by which quantum BCH codes can be constructed from classical codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes.

We determine the dimension and in some cases the minimum distance of primitive, narrow sense BCH codes of length n with small designed distance. We show that such a code contains its Euclidean dual code and, when the size of the field is a perfect square, also its Hermitian dual code. We establish two series of quantum error-correcting codes.

We prove quantum Hamming bound for stabilizer codes of minimum distance d = 5. Also, we compute the maximum length of single and double MDS stabilizer codes over finite fields. 1 Bounds on Quantum Codes It is desirable to study upper and lower bounds on the minimum distance and dimensions of quantum codes, so the computer search on the code parameter can be minimized and optimal codes can be known. It is a well-known fact that Singleton and Hamming bounds hold for classical codes [9]. We need some bounds on the achievable minimum distance of a quantum stabilizer code. Perhaps the simplest one is the Knill-LaFlamme bound, also called the quantum Singleton bound. The binary version of the quantum Singleton bound was first proved by Knill and Laflamme in [12], see also [1,2], and later generalized by Rains using weight enumerators in [16]. Theorem 1 (Quantum Singleton Bound). An ((n, K, d))q stabilizer code with K> 1 satisfies K ≤ q n−2d+2. Codes which meet the quantum Singleton bound are called quantum MDS codes. In [11] It has been shown that these codes cannot be indefinitely long and showed that the maximal length of a q-ary quantum MDS codes is upper bounded by 2q 2 − 2. This could probably be tightened to q 2 + 2. It would be interesting to find quantum MDS codes of length greater than q 2 + 2 since it would disprove the MDS Conjecture for classical codes [9]. A related open question is regarding the construction of codes with lengths between q and q 2 −1. At the moment there are no analytical methods for constructing a quantum MDS code of arbitrary length in this range (see [8] for some numerical results). Another important bound for quantum codes is the quantum Hamming bound. The quantum Hamming bound states (see [5, 6]) that: 1 Theorem 2 (Quantum Hamming Bound). Any pure ((n, K, d))q stabilizer code satisfies ⌊(d−1)/2⌋

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