Bounds on the minimum distance of additive quantum codes
Bounds on [[112,4]]2
lower bound: | 22 |
upper bound: | 37 |
Construction
Construction of a [[112,4,22]] quantum code:
[1]: [[109, 1, 25]] quantum code over GF(2^2)
cyclic code of length 109 with generating polynomial x^55 + w^2*x^54 + w*x^53 + w^2*x^52 + w^2*x^51 + w^2*x^50 + w*x^48 + x^47 + w*x^46 + x^44 + w*x^42 + x^41 + w^2*x^40 + x^39 + w*x^38 + x^36 + x^35 + w*x^33 + x^32 + w^2*x^31 + w^2*x^30 + x^29 + x^26 + w^2*x^25 + w^2*x^24 + x^23 + w*x^22 + x^20 + x^19 + w*x^17 + x^16 + w^2*x^15 + x^14 + w*x^13 + x^11 + w*x^9 + x^8 + w*x^7 + w^2*x^5 + w^2*x^4 + w^2*x^3 + w*x^2 + w^2*x + 1
[2]: [[106, 4, 22]] quantum code over GF(2^2)
Shortening of the stabilizer code of [1] at { 107 .. 109 }
[3]: [[111, 4, 22]] quantum code over GF(2^2)
ExtendCode [2] by 5
[4]: [[112, 4, 22]] quantum code over GF(2^2)
ExtendCode [3] by 1
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|0 0 0 0 1 0 0 1 1 1 0 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 0 1 1 0 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 1 0 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0]
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last modified: 2022-08-02
Notes
- All codes establishing the lower bounds where constructed using MAGMA.
- Most upper bounds on qubit codes for n≤100 are based on a MAGMA program by Eric Rains.
- For n>100, the upper bounds on qubit codes are weak (and not even monotone in k).
- Some additional information can be found in the book by Nebe, Rains, and Sloane.
- My apologies to all authors that have contributed codes to this table for not giving specific credits.
This page is maintained by
Markus Grassl
(codes@codetables.de).
Last change: 23.10.2014