1998 Summer Program: Coding and Cryptography Week 1: Codes, Design, and Cryptography

The paper is concerned with bounds for the quantum error correcting codes. Using the quantum MacWilliams identities, we generalize the linear programming approach from classical coding theory to the quantum case. Using this approach, we obtain Singleton type, Hamming type, and the first linear programming type bounds for quantum codes. We demonstrate that classical Singleton bound is valid for quantum error correcting codes despite the relaxation in the definition of the minimum distance of a quantum code. We also show that Hamming and first linear programming type bounds are also valid for quantum codes practically on all interval
0 < log K/n < 1.

Following are the slides used during the talk.

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Figure 1: VG is Varshamov-Gilbert; Sng is Singleton bound; LP1 and LP2are the linear
programming bounds; H is Hamming type bound; S is strengthening for linear codes.

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