Bounds on the minimum distance of additive quantum codes

Bounds on [[115,0]]₂

Construction

Construction of a [[115,0,25]] quantum code:
[1]:  [115, 58] Quasicyclic of degree 5 Linear Code over GF(2^2)
     QuasiCyclicCode of length 115 stacked to height  3 with generating polynomials: 1,  0,  w^2*x^10 + x^9 + x^7 + x^6 + w*x^5 + x^4 + w*x^3 + x^2 + w*x + w,  w*x^22 + x^21 + w^2*x^19 + x^18 + w*x^15 + w*x^13 + w^2*x^12 + w^2*x^11 + x^10 + x^9 + x^8 + w*x^7 + w*x^6 + w^2*x^5 + w*x^4 + x^3 + x^2 + w,  w^2*x^22 + w*x^21 + w^2*x^20 + x^18 + w^2*x^17 + w*x^16 + w*x^15 + w*x^13 + w^2*x^12 + w*x^11 + w^2*x^10 + w*x^9 + w^2*x^8 + x^7 + x^5 + x^4 + x^3 + w*x^2 + 1,  0,  1,  w*x^10 + x^9 + x^8 + x^6 + x^5 + w*x^4 + x^3 + w*x^2 + x + w^2,  w^2*x^22 + w^2*x^19 + x^17 + w*x^16 + x^15 + w^2*x^14 + w*x^13 + w*x^12 + w*x^10 + w^2*x^8 + w*x^7 + w*x^5 + x^4 + w^2*x^3 + w^2*x + 1,  x^22 + w*x^20 + x^19 + w*x^17 + x^16 + w*x^15 + w^2*x^13 + w^2*x^12 + w*x^10 + w*x^8 + w*x^7 + w*x^6 + w^2*x^5 + w^2*x^4 + w^2*x + 1,  0,  0,  x^11 + x^10 + x^6 + x^5 + x^4 + x^2 + 1,  x^22 + w^2*x^21 + x^20 + x^19 + w^2*x^18 + x^17 + w^2*x^16 + x^15 + x^14 + x^13 + w*x^12 + x^11 + w*x^10 + x^9 + w^2*x^8 + x^6 + x^5 + w*x^3 + w,  w*x^22 + x^21 + w^2*x^20 + w*x^18 + w^2*x^17 + w*x^16 + w*x^14 + x^13 + x^11 + w^2*x^9 + w^2*x^8 + x^6 + w*x^5 + x^4 + w^2*x^3 + w^2*x + 1
[2]:  [[116, 0, 26]] self-dual quantum code over GF(2^2)
     QuantumConstructionX applied to [1] with e = 1
[3]:  [[115, 0, 25]] self-dual quantum code over GF(2^2)
     Puncturing of [2] at { 116 }

    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 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 1 1 1 1 0 0 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 1 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 0 0]
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last modified: 2024-05-23

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.