Best Known (193, 193+54, s)-Nets in Base 4
(193, 193+54, 1539)-Net over F4 — Constructive and digital
Digital (193, 247, 1539)-net over F4, using
- 41 times duplication [i] based on digital (192, 246, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 82, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 82, 513)-net over F64, using
(193, 193+54, 4418)-Net over F4 — Digital
Digital (193, 247, 4418)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4247, 4418, F4, 54) (dual of [4418, 4171, 55]-code), using
- 310 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 9 times 0, 1, 30 times 0, 1, 65 times 0, 1, 93 times 0, 1, 106 times 0) [i] based on linear OA(4241, 4102, F4, 54) (dual of [4102, 3861, 55]-code), using
- construction X applied to Ce(53) ⊂ Ce(52) [i] based on
- linear OA(4241, 4096, F4, 54) (dual of [4096, 3855, 55]-code), using an extension Ce(53) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,53], and designed minimum distance d ≥ |I|+1 = 54 [i]
- linear OA(4235, 4096, F4, 53) (dual of [4096, 3861, 54]-code), using an extension Ce(52) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,52], and designed minimum distance d ≥ |I|+1 = 53 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(53) ⊂ Ce(52) [i] based on
- 310 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 9 times 0, 1, 30 times 0, 1, 65 times 0, 1, 93 times 0, 1, 106 times 0) [i] based on linear OA(4241, 4102, F4, 54) (dual of [4102, 3861, 55]-code), using
(193, 193+54, 1172226)-Net in Base 4 — Upper bound on s
There is no (193, 247, 1172227)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 51147 875575 378608 115928 866267 148794 702874 348448 798961 357116 051336 801895 930549 597003 995593 560999 212233 013582 677092 527723 008370 755249 822791 952581 920640 > 4247 [i]