Bounds on the minimum distance of additive quantum codes

Bounds on [[226,0]]₂

Construction

Construction of a [[226,0,22]] quantum code:
[1]:  [127, 64, 21] Cyclic Linear Code over GF(2)
     cyclic code of length 127 with generating polynomial x^63 + x^62 + x^61 + x^60 + x^58 + x^55 + x^50 + x^46 + x^44 + x^42 + x^40 + x^36 + x^35 + x^31 + x^29 + x^28 + x^27 + x^24 + x^22 + x^20 + x^19 + x^13 + x^11 + x^9 + x^4 + x^3 + x^2 + x + 1
[2]:  [128, 64, 22] Linear Code over GF(2)
     ExtendCode [1] by 1
[3]:  [[128, 0, 22]] CSS self-dual quantum code over GF(2^2)
     CSS code using  [2] and  [2]
[4]:  [[98, 0, 24]] self-dual quantum code over GF(2^2)
     QuasiCyclicCode of length 98 stacked to height  2 with generating polynomials: x^3 + x + 1,  w*x^44 + x^43 + w^2*x^42 + w^2*x^41 + w^2*x^40 + x^39 + w*x^37 + w*x^36 + w^2*x^35 + w^2*x^34 + w^2*x^32 + w^2*x^31 + x^30 + w^2*x^28 + w^2*x^27 + x^25 + w^2*x^22 + w^2*x^21 + x^19 + x^18 + w^2*x^17 + w^2*x^16 + w^2*x^13 + w*x^12 + x^11 + x^10 + w^2*x^9 + w*x^7 + x^5 + w*x^4 + w*x^3 + x^2 + w^2*x,  0,  x^46 + x^45 + x^44 + x^42 + x^39 + x^38 + x^37 + x^35 + x^32 + x^31 + x^30 + x^28 + x^25 + x^24 + x^23 + x^21 + x^18 + x^17 + x^16 + x^14 + x^11 + x^10 + x^9 + x^7 + x^4 + x^3 + x^2 + 1
[5]:  [[226, 0, 22]] self-dual quantum code over GF(2^2)
     direct sum of [4] and  [3]

    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 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 0 1 1 0 0 0 0 1 0 1 1 1 0 1 0 1 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 1 0 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 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 1 1 1 1 0 0 0 1 0 0 0 1 1 1 1 0 1 0 0 1 0 0 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
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last modified: 2019-05-16

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 necessarily 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.