Best Known (58−14, 58, s)-Nets in Base 4
(58−14, 58, 1028)-Net over F4 — Constructive and digital
Digital (44, 58, 1028)-net over F4, using
- 42 times duplication [i] based on digital (42, 56, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 14, 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, 14, 257)-net over F256, using
(58−14, 58, 1085)-Net over F4 — Digital
Digital (44, 58, 1085)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(458, 1085, F4, 14) (dual of [1085, 1027, 15]-code), using
- 45 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 5 times 0, 1, 11 times 0, 1, 22 times 0) [i] based on linear OA(452, 1034, F4, 14) (dual of [1034, 982, 15]-code), using
- construction XX applied to C1 = C([329,341]), C2 = C([331,342]), C3 = C1 + C2 = C([331,341]), and C∩ = C1 ∩ C2 = C([329,342]) [i] based on
- linear OA(446, 1023, F4, 13) (dual of [1023, 977, 14]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {329,330,…,341}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(446, 1023, F4, 12) (dual of [1023, 977, 13]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {331,332,…,342}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(451, 1023, F4, 14) (dual of [1023, 972, 15]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {329,330,…,342}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(441, 1023, F4, 11) (dual of [1023, 982, 12]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {331,332,…,341}, and designed minimum distance d ≥ |I|+1 = 12 [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([329,341]), C2 = C([331,342]), C3 = C1 + C2 = C([331,341]), and C∩ = C1 ∩ C2 = C([329,342]) [i] based on
- 45 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 5 times 0, 1, 11 times 0, 1, 22 times 0) [i] based on linear OA(452, 1034, F4, 14) (dual of [1034, 982, 15]-code), using
(58−14, 58, 109716)-Net in Base 4 — Upper bound on s
There is no (44, 58, 109717)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 83077 512851 557796 189779 850371 142800 > 458 [i]