@Book{Nielsen-book,
  author =	 {M. A. Nielsen and I. L. Chuang},
  title =	 {Quantum Computation and Quantum Infomation},
  publisher =	 {Cambridge Unive. Press},
  year =	 2000,
  address =	 {Cambridge, MA}
}

@Article{Calderbank-1997,
  title =	 {Quantum error correction via codes over {GF(4)}},
  author =	 {Calderbank, A. R. and Rains, E. M. and Shor,
                  P. M. and Sloane, N. J. A.},
  journal =	 {IEEE Trans. Info. Theory},
  year =	 1998,
  volume =	 44,
  pages =	 {1369-1387},
  abstract =	 {The problem of finding quantum error correcting
                  codes is transformed into the problem of finding
                  additive codes over the field GF(4) which are
                  self-orthogonal with respect to a certain trace
                  inner product. Many new codes and new bounds are
                  presented, as well as a table of upper and lower
                  bounds on such codes of length up to 30 qubits},
  doi =		 {10.1109/18.681315},
  url =		 {https://doi.org/10.1109/18.681315},
}

@PhdThesis{gottesman-thesis,
  author =	 {Daniel Gottesman},
  title =	 {Stabilizer Codes and Quantum Error Correction},
  school =	 {Caltech},
  year =	 1997,
  annote =	 {Controlling operational errors and decoherence is
                  one of the major challenges facing the field of
                  quantum computation and other attempts to create
                  specified many-particle entangled states. The field
                  of quantum error correction has developed to meet
                  this challenge. A group-theoretical structure and
                  associated subclass of quantum codes, the stabilizer
                  codes, has proved particularly fruitful in producing
                  codes and in understanding the structure of both
                  specific codes and classes of codes. I will give an
                  overview of the field of quantum error correction
                  and the formalism of stabilizer codes. In the
                  context of stabilizer codes, I will discuss a number
                  of known codes, the capacity of a quantum channel,
                  bounds on quantum codes, and fault-tolerant quantum
                  computation.},
  url =		 "https://arxiv.org/abs/quant-ph/9705052"
}

@ARTICLE{Ashikhmin-Knill-2001,
  author =	 {Ashikhmin, A. and Knill, E.},
  journal =	 {IEEE Trans. Info. Th.},
  title =	 {Nonbinary quantum stabilizer codes},
  year =	 2001,
  month =	 {Nov},
  volume =	 47,
  number =	 7,
  pages =	 {3065-3072},
  abstract =	 {We define and show how to construct nonbinary
                  quantum stabilizer codes. Our approach is based on
                  nonbinary error bases. It generalizes the
                  relationship between self-orthogonal codes over F4
                  and binary quantum codes to one between
                  self-orthogonal codes over F(q2 ) and q-ary quantum
                  codes for any prime power q},
  keywords =	 {Galois fields;binary codes;quantum
                  communication;binary quantum codes;coding
                  theory;encoding algorithms;nonbinary error
                  bases;nonbinary quantum stabilizer codes;q-ary
                  quantum codes;self-orthogonal codes;Algorithm design
                  and analysis;Code standards;Encoding;Fault
                  tolerance;Galois fields;Linear code;Quantum
                  computing;Quantum mechanics;Rain;Relativistic
                  quantum mechanics},
  doi =		 {10.1109/18.959288}
}

@ARTICLE{Ketkar-Klappenecker-Kumar-Sarvepalli-2006,
  author =	 {Ketkar, A. and Klappenecker, A. and Kumar, S. and
                  Sarvepalli, P. K.},
  journal =	 {{IEEE} Trans. Info. Th.},
  title =	 {Nonbinary Stabilizer Codes Over Finite Fields},
  year =	 2006,
  month =	 {Nov},
  volume =	 52,
  number =	 11,
  pages =	 {4892-4914},
  abstract =	 {One formidable difficulty in quantum communication
                  and computation is to protect information-carrying
                  quantum states against undesired interactions with
                  the environment. To address this difficulty, many
                  good quantum error-correcting codes have been
                  derived as binary stabilizer codes. Fault-tolerant
                  quantum computation prompted the study of nonbinary
                  quantum codes, but the theory of such codes is not
                  as advanced as that of binary quantum codes. This
                  paper describes the basic theory of stabilizer codes
                  over finite fields. The relation between stabilizer
                  codes and general quantum codes is clarified by
                  introducing a Galois theory for these objects. A
                  characterization of nonbinary stabilizer codes over
                  Fq in terms of classical codes over Fq 2 is provided
                  that generalizes the well-known notion of additive
                  codes over F4 of the binary case. This paper also
                  derives lower and upper bounds on the minimum
                  distance of stabilizer codes, gives several code
                  constructions, and derives numerous families of
                  stabilizer codes, including quantum Hamming codes,
                  quadratic residue codes, quantum Melas codes,
                  quantum Bose-Chaudhuri-Hocquenghem (BCH) codes, and
                  quantum character codes. The puncturing theory by
                  Rains is generalized to additive codes that are not
                  necessarily pure. Bounds on the maximal length of
                  maximum distance separable stabilizer codes are
                  given. A discussion of open problems concludes this
                  paper},
  keywords =	 {BCH codes;Galois fields;Hamming codes;error
                  correction codes;fault tolerant computing;quantum
                  communication;quantum computing;residue
                  codes;BCH;Bose-Chaudhuri-Hocquenghem code;Galois
                  theory;Hamming code;additive code;error-correcting
                  code;fault-tolerant quantum computation;finite
                  field;minimum distance;nonbinary stabilizer
                  code;puncturing theory;quadratic residue
                  code;quantum Melas code;quantum character
                  code;quantum communication;Computer science;Error
                  correction codes;Fault tolerance;Galois
                  fields;Information processing;Protection;Quantum
                  computing;Quantum mechanics;Rain;Upper
                  bound;Bose&#8211;Chaudhuri&#8211;Hocquenghem (BCH)
                  codes;MDS codes;Reed-Muller codes;bounds;nonbinary
                  codes;puncturing;quantum codes;self-orthogonal
                  codes},
  doi =		 {10.1109/TIT.2006.883612}, eprint={arXiv:quant-ph/0508070}
}

@Unpublished{Gottesman-prime-power-2014,
  author =	 {Daniel Gottesman},
  title =	 {Stabilizer codes with prime power qudits},
  note =	 {Invited talk at QEC 2014 (ETH Zurich)},
  url =
                  {https://www.qec14.ethz.ch/slides/DanielGottesman.pdf},
  year =	 2014,
  annote =	 {Quantum error correction of decoherence and faulty
                  control operations forms the backbone of all of
                  quantum information processing. Despite remarkable
                  progress on this front since the discovery of
                  quantum error correcting codes more than a decade
                  ago, important open problems in both theory and
                  applications to real physical systems remain. }
}

@article{Zeng-Pryadko-2018,
  title =	 {Higher-Dimensional Quantum Hypergraph-Product Codes
                  with Finite Rates},
  author =	 {Zeng, Weilei and Pryadko, Leonid P.},
  journal =	 {Phys. Rev. Lett.},
  volume =	 122,
  issue =	 23,
  pages =	 230501,
  numpages =	 6,
  year =	 2019,
  month =	 {Jun},
  publisher =	 {American Physical Society},
  doi =		 {10.1103/PhysRevLett.122.230501},
  eprint =	 {1810.01519},
  url =
                  {https://link.aps.org/doi/10.1103/PhysRevLett.122.230501}
}

@Article{Zeng-Pryadko-hprod-2020,
  title =	 {Minimal distances for certain quantum product codes
                  and tensor products of chain complexes},
  author =	 {Zeng, Weilei and Pryadko, Leonid P.},
  journal =	 {Phys. Rev. A},
  volume =	 102,
  issue =	 6,
  pages =	 062402,
  numpages =	 15,
  year =	 2020,
  doi =		 {10.1103/PhysRevA.102.062402},
  url =
                  {https://link.aps.org/doi/10.1103/PhysRevA.102.062402},
  eprint =	 {arXiv:2007.12152},
  annote =	 {We use a map to quantum error-correcting codes and a
                  subspace projection to get lower bounds for minimal
                  homological distances in a tensor product of two
                  chain complexes of vector spaces over a finite
                  field. Homology groups of such a complex are
                  described by the Künneth theorem. We give an
                  explicit expression for the distances when one of
                  the complexes is a linear map between two
                  spaces. The codes in the construction, subsystem
                  product codes and their gauge-fixed variants,
                  generalize several known families of quantum
                  error-correcting codes.}
}

@ARTICLE{Leon-1988,
  author =	 {Leon, J. S.},
  journal =	 {IEEE Trans. Info. Theory},
  title =	 {A probabilistic algorithm for computing minimum
                  weights of large error-correcting codes},
  year =	 1988,
  month =	 {Sep},
  volume =	 34,
  number =	 5,
  pages =	 {1354 -1359},
  abstract =	 {An algorithm is developed that can be used to find,
                  with a very low probability of error (10-100 or less
                  in many cases), the minimum weights of codes far too
                  large to be treated by any known exact
                  algorithm. The probabilistic method is used to find
                  minimum weights of all extended quadratic residue
                  codes of length 440 or less. The probabilistic
                  algorithm is presented for binary codes, but it can
                  be generalized to codes over GF(q) with q gt;2},
  keywords =	 {Binary codes;Computer errors;Computer science;Error
                  correction codes;Linear code;Mathematics;National
                  security;Statistics;error correction
                  codes;probability;binary codes;error-correcting
                  codes;minimum weights;probabilistic
                  algorithm;quadratic residue codes;},
  doi =		 {10.1109/18.21270},
  ISSN =	 {0018-9448},
}


@Article{Kruk-1989,
  author =	 {E. A. Kruk},
  title =	 {Decoding Complexity Bound for Linear Block Codes},
  journal =	 {Probl. Peredachi Inf.},
  year =	 1989,
  volume =	 25,
  number =	 3,
  pages =	 {103-107},
  note =	 {(In Russian)},
  url =		 {http://mi.mathnet.ru/eng/ppi665},
  annote =	 {A new complexity bound is derived for
                  maximum-likelihood decoding of linear block codes in
                  a memoryless q-ary symmetric channel. The bound is
                  the best among all known bounds in the entire range
                  of code rates. }
}

@ARTICLE{Coffey-Goodman-1990,
  author =	 {Coffey, J. T. and Goodman, R. M.},
  journal =	 {IEEE Trans. Info. Theory},
  title =	 {The complexity of information set decoding},
  year =	 {1990},
  month =	 {Sep},
  volume =	 {36},
  number =	 {5},
  pages =	 {1031 -1037},
  abstract =	 {Information set decoding is an algorithm for
                  decoding any linear code. Expressions for the
                  complexity of the procedure that are logarithmically
                  exact for virtually all codes are presented. The
                  expressions cover the cases of complete minimum
                  distance decoding and bounded hard-decision
                  decoding, as well as the important case of bounded
                  soft-decision decoding. It is demonstrated that
                  these results are vastly better than those for the
                  trivial algorithms of searching through all
                  codewords or through all syndromes, and are
                  significantly better than those for any other
                  general algorithm currently known. For codes over
                  large symbol fields, the procedure tends towards a
                  complexity that is subexponential in the symbol
                  size},
  keywords =	 {Decoding;Error correction;Helium;Linear
                  code;Polynomials;Redundancy;Vectors;decoding;bounded
                  hard-decision decoding;bounded soft-decision
                  decoding;complete minimum distance
                  decoding;complexity;information set decoding;linear
                  code;},
  doi =		 {10.1109/18.57202},
  ISSN =	 {0018-9448},
}

@Article{Cuellar-etal-2020,
  author =	 {M. P. Cu{\'e}llar and G{\'o}mez-Torrecillas, J.  and
                  F. J. Lobillo and G. Navarro},
  title =	 {Genetic algorithms with permutation-based
                  representation for computing the distance of linear
                  codes},
  year =	 2021,
  journal =	 {Swarm and Evolutionary Computation},
  volume =	 60,
  pages =	 100797,
  eprint =	 {arXiv:2002.12330},
  doi =		 {10.1016/j.swevo.2020.100797},
  annote =	 {Finding the minimum distance of linear codes is an
                  NP-hard problem. Traditionally, this computation has
                  been addressed by means of the design of algorithms
                  that find, by a clever exhaustive search, a linear
                  combination of some generating matrix rows that
                  provides a codeword with minimum weight. Therefore,
                  as the dimension of the code or the size of the
                  underlying finite field increase, so it does
                  exponentially the run time. In this work, we prove
                  that, given a generating matrix, there exists a
                  column permutation which leads to a reduced row
                  echelon form containing a row whose weight is the
                  code distance. This result enables the use of
                  permutations as representation scheme, in contrast
                  to the usual discrete representation, which makes
                  the search of the optimum polynomial time dependent
                  from the base field. In particular, we have
                  implemented genetic and CHC algorithms using this
                  representation as a proof of concept. Experimental
                  results have been carried out employing codes over
                  fields with two and eight elements, which suggests
                  that evolutionary algorithms with our proposed
                  permutation encoding are competitive with regard to
                  existing methods in the literature. As a by-product,
                  we have found and amended some inaccuracies in the
                  MAGMA Computational Algebra System concerning the
                  stored distances of some linear codes.}
}

@article{Dumer-Kovalev-Pryadko-bnd-2015,
  title =	 {Thresholds for correcting errors, erasures, and
                  faulty syndrome measurements in degenerate quantum
                  codes},
  author =	 {Dumer, I. and Kovalev, A. A. and Pryadko, L. P.},
  journal =	 {Phys. Rev. Lett.},
  volume =	 115,
  pages =	 050502,
  numpages =	 5,
  year =	 2015,
  publisher =	 {American Physical Society},
  doi =		 {10.1103/PhysRevLett.115.050502},
  url =
                  {https://doi.org/10.1103/PhysRevLett.115.050502},
  eprint =	 {1412.6172},
  annote =	 {We suggest a technique for constructing lower
                  (existence) bounds for the fault-tolerant threshold
                  to scalable quantum computation applicable to
                  degenerate quantum codes with sublinear distance
                  scaling. We give explicit analytic expressions
                  combining probabilities of erasures, depolarizing
                  errors, and phenomenological syndrome measurement
                  errors for quantum LDPC codes with logarithmic or
                  larger distances. These threshold estimates are
                  parametrically better than the existing analytical
                  bound based on percolation.  }
}

@ARTICLE{Dumer-Kovalev-Pryadko-IEEE-2017,
  author =	 {I. Dumer and A. A. Kovalev and L. P. Pryadko},
  journal =	 {IEEE Trans. Inf. Th.},
  title =	 {Distance Verification for Classical and Quantum {LDPC}
                  Codes},
  year =	 2017,
  volume =	 63,
  number =	 7,
  pages =	 {4675-4686},
  abstract =	 {The techniques of distance verification known for
                  general linear codes are first applied to the
                  quantum stabilizer codes. Then, these techniques are
                  considered for classical and quantum (stabilizer)
                  low-density-parity-check (LDPC) codes. New
                  complexity bounds for distance verification with
                  provable performance are derived using the average
                  weight spectra of the ensembles of LDPC codes. These
                  bounds are expressed in terms of the
                  erasure-correcting capacity of the corresponding
                  ensemble. We also present a new irreducible-cluster
                  technique that can be applied to any LDPC code and
                  takes advantage of parity-checks' sparsity for both
                  the classical and quantum LDPC codes. This technique
                  reduces complexity exponents of all existing
                  deterministic techniques designed for generic
                  stabilizer codes with small relative distances,
                  which also include all known families of the quantum
                  stabilizer LDPC codes.},
  keywords =	 {parity check codes;classical LDPC codes;complexity
                  exponents;distance verification;erasure-correcting
                  capacity;general linear codes;generic stabilizer
                  codes;irreducible-cluster technique;low-density
                  parity check codes;quantum LDPC codes;quantum
                  stabilizer codes;Algorithm design and
                  analysis;Complexity
                  theory;Decoding;Generators;Linear codes;Parity check
                  codes;Quantum computing;Distance verification;LDPC
                  codes;erasure correction;list decoding;quantum
                  stabilizer codes},
  doi =		 {10.1109/TIT.2017.2690381},
  ISSN =	 {0018-9448},
  month =	 {July},
}

@TechReport{Steel-1953,
  author =	 {R. G. D. Steel},
  title =	 {Relation Between {P}oisson and Multinomial
                  Distributions},
  institution =	 {Cornell University},
  year =	 1953,
  type =	 {Biometrics Unit Technical Reports},
  number =	 {BU-39-M},
  url =		 "https://ecommons.cornell.edu/handle/1813/32480",
  annote =	 {Relation between k trials on independent Poisson
                  populations with different means and the fixed total n, and a multinomual
                  distribution with k outcomes.}
}

@article{Chernoff-Lehmann-1954,
  fullauthor =	 {Herman Chernoff and E. L. Lehmann},
  title =	 {The Use of Maximum Likelihood Estimates in $\chi^2$
                  Tests for Goodness of Fit},
  year =	 1954,
  author =	 {H. Chernoff and E. L. Lehmann},
  journal =	 {The Annals of Mathematical Statistics},
  volume =	 25,
  number =	 3,
  publisher =	 {Institute of Mathematical Statistics},
  pages =	 {579 -- 586},
  abstract =	 {The usual test that a sample comes from a
                  distribution of given form is performed by counting
                  the number of observations falling into specified
                  cells and applying the $\chi^2$ test to these
                  frequencies. In estimating the parameters for this
                  test, one may use the maximum likelihood (or
                  equivalent) estimate based (1) on the cell
                  frequencies, or (2) on the original
                  observations. This paper shows that in (2), unlike
                  the well known result for (1), the test statistic
                  does not have a limiting $\chi^2$-distribution, but
                  that it is stochastically larger than would be
                  expected under the $\chi^2$ theory. The limiting
                  distribution is obtained and some examples are
                  computed. These indicate that the error is not
                  serious in the case of fitting a Poisson
                  distribution, but may be so for the fitting of a
                  normal.},
  doi =		 {10.1214/aoms/1177728726},
  URL =		 {https://doi.org/10.1214/aoms/1177728726}
}

@book{Cramer-book-1999,
  title =	 {Mathematical Methods of Statistics ({PMS}-9)},
  author =	 {Harald Cram{\'e}r},
  publisher =	 {Princeton University Press},
  year =	 1999,
  ISBN =	 9780691005478,
  URL =		 {https://www.jstor.org/stable/j.ctt1bpm9r4},
  abstract =	 {In this classic of statistical mathematical theory,
                  Harald Cramér joins the two major lines of
                  development in the field: while British and American
                  statisticians were developing the science of
                  statistical inference, French and Russian
                  probabilitists transformed the classical calculus of
                  probability into a rigorous and pure mathematical
                  theory. The result of Cramér's work is a masterly
                  exposition of the mathematical methods of modern
                  statistics that set the standard that others have
                  since sought to follow.For anyone with a working
                  knowledge of undergraduate mathematics the book is
                  self contained. The first part is an introduction to
                  the fundamental concept of a distribution and of
                  integration with respect to a distribution. The
                  second part contains the general theory of random
                  variables and probability distributions while the
                  third is devoted to the theory of sampling,
                  statistical estimation, and tests of significance.}
}


@misc{ConwayPol-2022,
  author =	 {Frank L{\"u}beck},
  title =	 {Conway polynomials for finite fields},
  url =
                  {https://www.math.rwth-aachen.de/~Frank.Luebeck/data/ConwayPol/},
  note =	 {[Downloaded on 2022-02-19]},
  year =	 2021,
  annote =	 {Extensive list of Conway polynomials used to define
                  extension Galois fields.}
}

