Bounds on the minimum distance of additive quantum codes

Bounds on [[238,196]]₂

Construction

Construction of a [[238,196,7]] quantum code:
[1]:  [[237, 198, 7]] quantum code over GF(2^2)
     QuasiCyclicCode of length 237 with generating polynomials: w*x^78 + w*x^77 + w*x^75 + w*x^72 + w*x^70 + w*x^69 + w*x^68 + w*x^67 + w*x^66 + w*x^63 + w*x^62 + w*x^58 + w*x^57 + w*x^55 + w*x^54 + w*x^53 + w*x^50 + w*x^48 + w*x^44 + w*x^42 + w*x^41 + w^2*x^40 + w^2*x^39 + w^2*x^37 + w*x^36 + w^2*x^35 + w^2*x^32 + w*x^31 + x^29 + w^2*x^28 + w*x^27 + w*x^26 + w*x^25 + x^24 + w*x^23 + x^22 + w*x^21 + w*x^19 + w^2*x^18 + w^2*x^17 + x^16 + x^15 + w*x^14 + w^2*x^13 + w^2*x^12 + w^2*x^11 + w*x^9 + w*x^8 + x^6 + w^2*x^4 + x^3 + 1,  w^2*x^78 + w^2*x^77 + w^2*x^75 + w^2*x^73 + w^2*x^72 + w^2*x^71 + w^2*x^70 + x^69 + x^66 + x^65 + x^64 + w*x^63 + w^2*x^60 + x^59 + x^58 + x^57 + x^56 + w^2*x^53 + w^2*x^52 + w*x^51 + x^48 + w*x^47 + w^2*x^46 + w*x^45 + w*x^43 + w*x^42 + w*x^40 + w^2*x^39 + w^2*x^38 + x^37 + w^2*x^36 + w*x^35 + w^2*x^32 + x^31 + x^30 + w*x^29 + w^2*x^28 + w*x^27 + w^2*x^26 + w*x^24 + x^23 + x^21 + w*x^20 + w*x^18 + w*x^17 + x^16 + w^2*x^15 + x^13 + w^2*x^12 + w^2*x^11 + w^2*x^10 + w*x^8 + w*x^6 + x^5 + w^2*x^4 + w*x^3 + x + w,  x^76 + w*x^75 + w*x^74 + w*x^73 + w^2*x^72 + x^71 + x^69 + w^2*x^68 + x^67 + w*x^66 + w^2*x^65 + x^64 + w^2*x^62 + w*x^61 + x^60 + x^59 + w*x^57 + x^56 + x^55 + x^54 + w*x^51 + w^2*x^50 + w*x^49 + w*x^48 + w^2*x^47 + w^2*x^45 + w*x^44 + x^43 + x^42 + x^41 + w*x^40 + w*x^39 + x^38 + w*x^37 + x^36 + w*x^35 + w*x^34 + w*x^33 + w*x^32 + w*x^31 + w*x^30 + x^29 + w^2*x^28 + w*x^27 + x^26 + w^2*x^25 + w^2*x^24 + w*x^23 + x^21 + w*x^20 + w*x^19 + x^17 + w*x^16 + x^14 + w^2*x^12 + w*x^11 + w^2*x^9 + w*x^7 + w^2*x^4 + w^2*x^3 + x + 1
[2]:  [[237, 196, 7]] quantum code over GF(2^2)
     Subcode of [1]
[3]:  [[238, 196, 7]] quantum code over GF(2^2)
     ExtendCode [2] by 1

    stabilizer matrix:

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last modified: 2024-05-12

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.