Asymptotically Good Quantum Codes
Using algebraic geometry codes we give a polynomial construction of quantum codes with asymptotically non-zero rate and relative distance.
quant-ph↗
arXiv subjects
S. Litsyn
Publications and source records attributed to S. Litsyn.
Polynomial method in coding and information theory
Polynomial, or Delsarte's, method in coding theory accounts for a variety of structural results on, and bounds on the size of, extremal configurations (codes and designs) in various metric spaces. In recent works of the authors the applicability of the method was extended to cover a wider range of problems in coding and information theory. In this paper we present a general framework for the method which includes previous results as particular cases. We explain how this generalization leads to new asymptotic bounds on the performance of codes in binary-input memoryless channels and the Gaussian channel, which improve the results of Shannon et al. of 1959-67, and to a number of other results in combinatorial coding theory.
math.CO↗
Quantum Error Detection II: Bounds
In Part II we show that there exist quantum codes whose probability of undetected error falls exponentially with the length of the code and derive bounds on this exponent.The lower (existence) bound for stabilizer codes is proved by a counting argument for classical self-orthogonal quaternary codes. Upper bounds for any quantum codes are proved by linear programming. We present two general solutions of the LP problem. Together they give an upper bound on the exponent of undetected error. The upper and lower asymptotic bounds coincide for a certain interval of code rates close to 1.
quant-ph↗