Best Known (122, 155, s)-Nets in Base 4
(122, 155, 1049)-Net over F4 — Constructive and digital
Digital (122, 155, 1049)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 23, 21)-net over F4, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 7 and N(F) ≥ 21, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- digital (99, 132, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 33, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 33, 257)-net over F256, using
- digital (7, 23, 21)-net over F4, using
(122, 155, 4029)-Net over F4 — Digital
Digital (122, 155, 4029)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4155, 4029, F4, 33) (dual of [4029, 3874, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4155, 4131, F4, 33) (dual of [4131, 3976, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- linear OA(4145, 4097, F4, 33) (dual of [4097, 3952, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4121, 4097, F4, 27) (dual of [4097, 3976, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(410, 34, F4, 5) (dual of [34, 24, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4155, 4131, F4, 33) (dual of [4131, 3976, 34]-code), using
(122, 155, 1413321)-Net in Base 4 — Upper bound on s
There is no (122, 155, 1413322)-net in base 4, because
- 1 times m-reduction [i] would yield (122, 154, 1413322)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 521 484598 223180 588289 144186 352928 493640 045507 559400 913297 697510 273385 326432 176789 597346 814172 > 4154 [i]