Best Known (198−43, 198, s)-Nets in Base 4
(198−43, 198, 1052)-Net over F4 — Constructive and digital
Digital (155, 198, 1052)-net over F4, using
- 42 times duplication [i] based on digital (153, 196, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 49, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
- trace code for nets [i] based on digital (6, 49, 263)-net over F256, using
(198−43, 198, 4135)-Net over F4 — Digital
Digital (155, 198, 4135)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4198, 4135, F4, 43) (dual of [4135, 3937, 44]-code), using
- 21 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 12 times 0) [i] based on linear OA(4194, 4110, F4, 43) (dual of [4110, 3916, 44]-code), using
- construction X applied to C([0,21]) ⊂ C([0,20]) [i] based on
- linear OA(4193, 4097, F4, 43) (dual of [4097, 3904, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(4181, 4097, F4, 41) (dual of [4097, 3916, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 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
- construction X applied to C([0,21]) ⊂ C([0,20]) [i] based on
- 21 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 12 times 0) [i] based on linear OA(4194, 4110, F4, 43) (dual of [4110, 3916, 44]-code), using
(198−43, 198, 1286061)-Net in Base 4 — Upper bound on s
There is no (155, 198, 1286062)-net in base 4, because
- 1 times m-reduction [i] would yield (155, 197, 1286062)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 40348 093649 715411 569561 398000 555663 301902 156491 980470 699203 362772 416373 375530 956458 483104 810805 665434 531596 312569 668162 > 4197 [i]