Best Known (244−44, 244, s)-Nets in Base 4
(244−44, 244, 1556)-Net over F4 — Constructive and digital
Digital (200, 244, 1556)-net over F4, using
- 41 times duplication [i] based on digital (199, 243, 1556)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 27, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- digital (172, 216, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 72, 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, 72, 513)-net over F64, using
- digital (5, 27, 17)-net over F4, using
- (u, u+v)-construction [i] based on
(244−44, 244, 16433)-Net over F4 — Digital
Digital (200, 244, 16433)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4244, 16433, F4, 44) (dual of [16433, 16189, 45]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4241, 16428, F4, 44) (dual of [16428, 16187, 45]-code), using
- construction X applied to Ce(44) ⊂ Ce(37) [i] based on
- linear OA(4232, 16384, F4, 45) (dual of [16384, 16152, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(4197, 16384, F4, 38) (dual of [16384, 16187, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(49, 44, F4, 5) (dual of [44, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- a “DaH†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- construction X applied to Ce(44) ⊂ Ce(37) [i] based on
- linear OA(4241, 16430, F4, 42) (dual of [16430, 16189, 43]-code), using Gilbert–Varšamov bound and bm = 4241 > Vbs−1(k−1) = 71823 556055 889125 112864 785520 892465 926082 006627 317572 728102 538475 764248 876948 323668 917514 216900 423327 644418 103240 787918 658475 518238 264807 077512 [i]
- linear OA(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(4241, 16428, F4, 44) (dual of [16428, 16187, 45]-code), using
- construction X with Varšamov bound [i] based on
(244−44, 244, large)-Net in Base 4 — Upper bound on s
There is no (200, 244, large)-net in base 4, because
- 42 times m-reduction [i] would yield (200, 202, large)-net in base 4, but