Linear programming bounds for entanglement-assisted quantum error-correcting codes by split weight enumerators

Ching-Yi Lai, Alexei Ashikhmin

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


Linear programming approaches have been applied to derive upper bounds on the size of classical and quantum codes. In this paper, we derive similar results for general quantum codes with entanglement assistance by considering a type of split weight enumerator. After deriving the MacWilliams identities for these enumerators, we are able to prove algebraic linear programming bounds, such as the Singleton bound, the Hamming bound, and the first linear programming bound. Our Singleton bound and Hamming bound are more general than the previous bounds for entanglement-assisted quantum stabilizer codes. In addition, we show that the first linear programming bound improves the Hamming bound when the relative distance is sufficiently large. On the other hand, we obtain additional constraints on the size of Pauli subgroups for quantum codes, which allow us to improve the linear programming bounds on the minimum distance of quantum codes of small length. In particular, we show that there is no [[27, 15, 5]] or [[28, 14, 6]] stabilizer code. We also discuss the existence of some entanglement-assisted quantum stabilizer codes with maximal entanglement. As a result, the upper and lower bounds on the minimum distance of maximal-entanglement quantum stabilizer codes with length up to 20 are significantly improved.

Original languageEnglish
Pages (from-to)622-639
Number of pages18
JournalIEEE Transactions on Information Theory
Issue number1
StatePublished - 1 Jan 2018


  • Entanglement-assisted quantum codes
  • Linear programming bounds
  • MacWilliams identities
  • Split weight enumerators
  • Stabilizer codes

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