Bounds on the minimum distance of additive quantum codes

Bounds on [[220,172]]₂

Construction

Construction of a [[220,172,8]] quantum code:
[1]:  [[220, 174, 8]] quantum code over GF(2^2)
     QuasiTwistedCyclicCode of length 220 and constant w with generators: x^32 + w^2*x^31 + x^30 + w^2*x^28 + x^27 + w^2*x^25 + x^24 + x^23 + x^17 + x^16 + w^2*x^14 + w*x^13 + w^2*x^12 + w*x^11 + w^2*x^10 + w^2*x^9 + x^8 + x^7 + w*x^5 + x^4 + x^3 + x + w^2,  x^54 + x^52 + x^51 + x^50 + w*x^49 + w^2*x^48 + w*x^47 + w*x^46 + w^2*x^45 + x^44 + w*x^42 + x^40 + w^2*x^37 + x^36 + w^2*x^35 + x^34 + x^33 + w*x^32 + w*x^29 + x^27 + w^2*x^26 + x^25 + w*x^24 + x^23 + w*x^22 + w*x^21 + x^20 + x^19 + w*x^18 + w*x^17 + x^16 + x^15 + w*x^14 + w^2*x^12 + w^2*x^10 + w*x^9 + x^8 + w*x^7 + x^6 + x^5 + x^4 + x + w,  x^54 + w*x^53 + w*x^52 + w^2*x^50 + w^2*x^49 + w*x^47 + w^2*x^46 + x^41 + w^2*x^38 + x^35 + w^2*x^34 + x^33 + w^2*x^31 + w^2*x^28 + w*x^27 + w^2*x^26 + w^2*x^25 + x^24 + w*x^22 + w^2*x^21 + x^19 + w^2*x^18 + x^17 + w*x^16 + w^2*x^14 + w^2*x^13 + w*x^12 + w*x^11 + w*x^9 + w^2*x^7 + w*x^6 + w^2*x + 1,  x^54 + w^2*x^53 + w^2*x^51 + w*x^49 + w^2*x^48 + x^46 + w^2*x^44 + x^43 + w^2*x^42 + w^2*x^41 + x^39 + w^2*x^37 + w^2*x^36 + w*x^35 + w^2*x^34 + x^32 + x^31 + w^2*x^30 + x^29 + w^2*x^28 + w*x^27 + x^26 + w^2*x^25 + w^2*x^24 + w*x^22 + x^21 + w*x^20 + x^19 + x^18 + w*x^15 + x^14 + w*x^13 + w*x^11 + x^10 + x^9 + x^7 + w^2*x^6 + w^2*x^5 + w*x^3 + x + w^2
[2]:  [[220, 172, 8]] quantum code over GF(2^2)
     Subcode of [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 1 0 1 1 1 0 1 1 0 0 1 0 1 0 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 0 0 1 1 1 0 0 1 0 0 1 0 0 0 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 0 0 1 0 1 1 0 1 0 1 0 0 0 1 0 0 1 1 1 0 1 0 1 1 0 0 1 1 1 1 1 1 1 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 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 0 0 0 1 0 0 1 1 1 1 1 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 1 0 0 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 1 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 1 1 0 1 1 0 0 1 1]
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last modified: 2024-08-22

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.