Best Known (198−35, 198, s)-Nets in Base 4
(198−35, 198, 1539)-Net over F4 — Constructive and digital
Digital (163, 198, 1539)-net over F4, using
- t-expansion [i] based on digital (162, 198, 1539)-net over F4, using
- 3 times m-reduction [i] based on digital (162, 201, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 67, 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, 67, 513)-net over F64, using
- 3 times m-reduction [i] based on digital (162, 201, 1539)-net over F4, using
(198−35, 198, 16443)-Net over F4 — Digital
Digital (163, 198, 16443)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4198, 16443, F4, 35) (dual of [16443, 16245, 36]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4197, 16441, F4, 35) (dual of [16441, 16244, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,13]) [i] based on
- linear OA(4183, 16385, F4, 35) (dual of [16385, 16202, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(4141, 16385, F4, 27) (dual of [16385, 16244, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(414, 56, F4, 7) (dual of [56, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- a “GraXX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- construction X applied to C([0,17]) ⊂ C([0,13]) [i] based on
- linear OA(4197, 16442, F4, 34) (dual of [16442, 16245, 35]-code), using Gilbert–Varšamov bound and bm = 4197 > Vbs−1(k−1) = 8284 799817 232589 076142 652719 725747 424571 602237 512174 101808 593468 927559 714963 585715 799578 017076 611245 528399 254105 363636 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 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
- linear OA(4197, 16441, F4, 35) (dual of [16441, 16244, 36]-code), using
- construction X with Varšamov bound [i] based on
(198−35, 198, large)-Net in Base 4 — Upper bound on s
There is no (163, 198, large)-net in base 4, because
- 33 times m-reduction [i] would yield (163, 165, large)-net in base 4, but