Bounds on the minimum distance of additive quantum codes

Bounds on [[238,189]]₂

Construction

Construction of a [[238,189,8]] quantum code:
[1]:  [[235, 189, 8]] quantum code over GF(2^2)
     QuasiCyclicCode of length 235 with generating polynomials: x^24 + x^23 + x^20 + x^18 + x^15 + x^12 + x^11 + x^9 + x^8 + x^5 + x^4 + 1,  w^2*x^46 + w^2*x^45 + x^44 + w*x^43 + w^2*x^41 + w^2*x^39 + w^2*x^38 + x^37 + x^36 + x^32 + x^31 + x^30 + w^2*x^28 + x^26 + x^24 + w^2*x^23 + w*x^21 + w*x^20 + x^19 + w*x^18 + w^2*x^17 + x^16 + x^15 + w*x^14 + x^12 + w^2*x^9 + x^8 + w*x^7 + x^6 + x^5 + w^2*x^3 + x^2 + x + 1,  w*x^46 + w*x^45 + x^43 + w*x^41 + w*x^40 + x^39 + w*x^38 + w*x^37 + w*x^35 + w^2*x^34 + w^2*x^32 + w*x^30 + x^29 + w^2*x^28 + x^26 + w*x^23 + w*x^20 + w*x^19 + x^16 + x^13 + w*x^12 + w*x^11 + x^9 + x^8 + x^7 + w^2*x^5 + w^2*x^4 + w^2*x^3 + x^2 + w*x,  x^46 + w^2*x^44 + w*x^42 + w^2*x^41 + w^2*x^38 + w*x^35 + w^2*x^33 + w^2*x^32 + w*x^31 + w^2*x^30 + x^29 + x^28 + w^2*x^27 + x^25 + w^2*x^24 + w^2*x^23 + w*x^22 + w^2*x^21 + x^20 + x^19 + x^18 + w*x^17 + w*x^15 + x^14 + x^12 + w*x^11 + w*x^10 + x^9 + w*x^6 + w^2*x^5 + w*x^3 + w^2*x^2 + w^2*x + w^2,  w^2*x^46 + w^2*x^45 + w*x^43 + x^42 + w^2*x^41 + w^2*x^40 + w*x^34 + w^2*x^33 + w*x^32 + w^2*x^31 + x^30 + w^2*x^29 + w^2*x^28 + w*x^27 + w*x^26 + w^2*x^25 + w*x^24 + w*x^23 + x^22 + w*x^21 + x^20 + w*x^19 + x^16 + x^15 + w^2*x^14 + x^13 + w^2*x^12 + w^2*x^10 + w^2*x^9 + w*x^8 + w^2*x^7 + x^6 + w^2*x^5 + w^2*x^4 + w*x^2 + w
[2]:  [[238, 189, 8]] quantum code over GF(2^2)
     ExtendCode [1] by 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 1 0 0 0 1 1 0 0 1 1 0 1 1 0 0 1 0 0 1 0 1 0 0 1 1 0 1 0 1 0 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 1 1 0 1 0 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 1 0 0 0 0 1 0 0 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 0 1 0 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 0 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 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 0 0 0 1 0 1 0 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 1 1 0 1 1 0 0 0 1 1 0 0 0 1 0 1 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 1 0 1 1 1 1 0 1 0 1 0 0 0 0 1 0 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.