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...in the Quantum case, because the probability of finding two sparse row-orthogonal matrices is extremely small. A number of constructions have been suggested by classical coding theorists nevertheless =-=[15, 1, 2, 8, 13, 20]-=- but they do not produce families of Quantum LDPC codes with a minimum distance growing with the blocklength. While this may be tolerable for practical constructions of fixed size, this is clearly an ...

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...t the design of high performance quantum LDPC codes is much more complicated than in the classical setting. In particular, most constructions suggested in the literature [4], [3], [5], [6], [7], [8], =-=[9]-=-, [10], [11] suffer from having either a bounded minimum distance or a vanishing rate. There are only a few exceptions, namely [12], [13], [14], [15]. However in all these constructions, unlike in the...

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...odes can be used to derive classes of symmetric quantum codes [4], [15] and asymmetric quantum codes [8], [20]. The literature lacks many construction of algebraic quantum LDPC codes, see for example =-=[1]-=-, [15] and references therein. II. CONSTRUCTING LDPC CODES Let Fq denote a finite field of characteristic p with q elements. Recall that the set F∗ q = Fq \ {0} of nonzero field elements is a multipli...

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...al-based construction of quasi-cyclic CSS QLDPC code by Hagiwara and Imai [23], classical quasi-cyclic LDPCbased construction of QLDPC codes by Hsieh et al. [24], finite geometric construction by Aly =-=[25]-=-, and BCH- and finite geometry LDPC code-based asymmetric QLDPC codes construction by Sarvepalli et al. [26]. It is remarkable that the error-correcting capability of some of these QLDPC codes are bet...

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...so well in the classical setting are pointless here. There have been a number of attempts at overcoming this difficulty and a variety of methods for constructing quantum LDPC codes have been proposed =-=[19, 16, 18, 5, 6, 17, 12, 13, 8, 21, 1, 2, 14, 23]-=-. However, all of these constructions suffer from disappointingly small minimum distances, namely whenever they have non-vanishing rate and parity-check matrices with bounded row-weight, their minimum...

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...II TYPE-I STABILIZER CODES OF LENGTH N = 4n− 1 FOR n ≤ 25. d† IS THE MINIMUM WEIGHT OF OPERATOR E ∈ S . n [[N,K, dmin]] d † 2 [[7, 3, 2]] 4 6 [[23, 11, 2]] 8 8 [[31, 15, 2]] 8 12 [[47, 23, 2]] 12 18 [=-=[71, 35, 2]-=-] 16 20 [[79, 39, 2]] 16 37 dmi n N K N [[136,15,12]] 2n-1 6 Fig. 2. Known asymptotic quantum coding bounds and some designed Type-I and Type-II stabilizer codes. VI. CONCLUSION AND DISCUSSIONS In thi...

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... for any pair (i, j) of adjacent vertices. Example 14 (LDPC codes) Any LDPC code whose paritycheck matrix contains orthogonal rows can be used to construct a normalizer label code C⊥, see e.g. [5], =-=[15]-=-, [16] and references therein. In terms of FFGs, such LDPC codes are expressed with the help of equal and single-parity-check function nodes. The FFG of the corresponding stabilizer label code C can a...

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...m LDPC, or QLDPC, codes were suggested by Postol in 2001 [13], and many examples were constructed by MacKay et al. [14]. Since then, many other classes of QLDPC codes have been constructed (see [15], =-=[16]-=- and references therin). However, many of these codes have unknown or small minimum distance. Tillich and Zemor [15] constructed fixedrate QLDPC codes with minimum distances that increase as the squar...