Best Known (23−6, 23, s)-Nets in Base 4
(23−6, 23, 514)-Net over F4 — Constructive and digital
Digital (17, 23, 514)-net over F4, using
- base reduction for projective spaces (embedding PG(11,16) in PG(22,4)) for nets [i] based on digital (6, 12, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 6, 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, 6, 257)-net over F256, using
(23−6, 23, 1039)-Net over F4 — Digital
Digital (17, 23, 1039)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(423, 1039, F4, 6) (dual of [1039, 1016, 7]-code), using
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(422, 1034, F4, 6) (dual of [1034, 1012, 7]-code), using
- construction XX applied to C1 = C([337,341]), C2 = C([339,342]), C3 = C1 + C2 = C([339,341]), and C∩ = C1 ∩ C2 = C([337,342]) [i] based on
- linear OA(416, 1023, F4, 5) (dual of [1023, 1007, 6]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {337,338,339,340,341}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(416, 1023, F4, 4) (dual of [1023, 1007, 5]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {339,340,341,342}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(421, 1023, F4, 6) (dual of [1023, 1002, 7]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {337,338,…,342}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(411, 1023, F4, 3) (dual of [1023, 1012, 4]-code or 1023-cap in PG(10,4)), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {339,340,341}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(41, 6, F4, 1) (dual of [6, 5, 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(40, 5, F4, 0) (dual of [5, 5, 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 XX applied to C1 = C([337,341]), C2 = C([339,342]), C3 = C1 + C2 = C([339,341]), and C∩ = C1 ∩ C2 = C([337,342]) [i] based on
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(422, 1034, F4, 6) (dual of [1034, 1012, 7]-code), using
(23−6, 23, 25004)-Net in Base 4 — Upper bound on s
There is no (17, 23, 25005)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 70 371578 476396 > 423 [i]