Bounds on the minimum distance of additive quantum codes

Bounds on [[197,0]]₂

Construction

Construction of a [[197,0,30]] quantum code:
[1]:  [199, 100, 31] Cyclic Linear Code over GF(2)
     cyclic code of length 199 with generating polynomial x^99 + x^97 + x^96 + x^94 + x^91 + x^90 + x^89 + x^88 + x^87 + x^83 + x^79 + x^77 + x^76 + x^75 + x^71 + x^69 + x^67 + x^66 + x^59 + x^57 + x^56 + x^55 + x^54 + x^52 + x^49 + x^48 + x^45 + x^43 + x^42 + x^41 + x^40 + x^37 + x^35 + x^33 + x^32 + x^30 + x^29 + x^27 + x^23 + x^22 + x^21 + x^17 + x^14 + x^12 + x^10 + x^8 + x^6 + x^5 + x^3 + x + 1
[2]:  [200, 100, 32] Linear Code over GF(2)
     ExtendCode [1] by 1
[3]:  [[200, 0, 32]] CSS self-dual quantum code over GF(2^2)
     CSS code using  [2] and  [2]
[4]:  [[198, 0, 31]] CSS self-dual quantum code over GF(2^2)
     shortening & puncturing CSS code  [3]
[5]:  [[197, 0, 30]] self-dual quantum code over GF(2^2)
     Puncturing of [4] at { 198 }

    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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 0 1 1 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 0 1 1 0 1 0 1 0 0 1 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 1 1 0 1 0 0 0 1 0 0 0 1 1 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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: 2022-08-08

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.