Bounds on the minimum distance of additive quantum codes

Bounds on [[239,0]]₂

Construction

Construction of a [[239,0,20]] quantum code:
[1]:  [[127, 0, 21]] self-dual quantum code over GF(2^2)
     QuasiCyclicCode of length 127 stacked to height 2 with generating polynomials: x^126 + x^125 + x^124 + x^122 + x^119 + x^114 + x^110 + x^108 + x^106 + x^104 + x^100 + x^99 + x^95 + x^93 + x^92 + x^91 + x^88 + x^86 + x^84 + x^83 + x^77 + x^75 + x^73 + x^68 + x^67 + x^66 + x^65 + x^64 + 1,  w*x^126 + w*x^122 + w*x^121 + w*x^118 + w*x^117 + w*x^116 + w*x^111 + w*x^109 + w*x^108 + w*x^107 + w*x^106 + w*x^105 + w*x^99 + w*x^98 + w*x^95 + w*x^93 + w*x^92 + w*x^89 + w*x^88 + w*x^87 + w*x^83 + w*x^81 + w*x^77 + w*x^74 + w*x^72 + w*x^70 + w*x^67 + w*x^64 + w*x^63 + w
[2]:  [111, 57, 25] Constacyclic by w Linear Code over GF(2^2)
     constacyclic code generated by x^54 + w^2*x^53 + x^52 + x^51 + x^50 + w*x^49 + w^2*x^48 + w^2*x^47 + w*x^46 + x^45 + w^2*x^43 + x^42 + w^2*x^40 + w*x^37 + x^36 + x^35 + w^2*x^31 + x^28 + x^27 + w*x^25 + x^24 + x^23 + w^2*x^20 + w*x^19 + w^2*x^17 + w^2*x^16 + x^14 + w*x^13 + w^2*x^11 + x^10 + w^2*x^7 + w^2*x^3 + w*x^2 + w*x + 1 with shift constant w
[3]:  [[114, 0, 26]] self-dual quantum code over GF(2^2)
     QuantumConstructionX applied to [2] with e = 3
[4]:  [[112, 0, 24]] self-dual quantum code over GF(2^2)
     Puncturing of [3] at { 113 .. 114 }
[5]:  [[239, 0, 20]] self-dual quantum code over GF(2^2)
     direct sum of [4] and  [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 1 1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 1 1 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 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 0 1 0 1 1 0 1 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.