Best Known (109−27, 109, s)-Nets in Base 4
(109−27, 109, 1028)-Net over F4 — Constructive and digital
Digital (82, 109, 1028)-net over F4, using
- 41 times duplication [i] based on digital (81, 108, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 27, 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, 27, 257)-net over F256, using
(109−27, 109, 1194)-Net over F4 — Digital
Digital (82, 109, 1194)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4109, 1194, F4, 27) (dual of [1194, 1085, 28]-code), using
- 153 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 16 times 0, 1, 27 times 0, 1, 40 times 0, 1, 51 times 0) [i] based on linear OA(4101, 1033, F4, 27) (dual of [1033, 932, 28]-code), using
- construction XX applied to C1 = C([1022,24]), C2 = C([0,25]), C3 = C1 + C2 = C([0,24]), and C∩ = C1 ∩ C2 = C([1022,25]) [i] based on
- linear OA(496, 1023, F4, 26) (dual of [1023, 927, 27]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,24}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(496, 1023, F4, 26) (dual of [1023, 927, 27]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4101, 1023, F4, 27) (dual of [1023, 922, 28]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,25}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(491, 1023, F4, 25) (dual of [1023, 932, 26]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,24], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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
- linear OA(40, 5, F4, 0) (dual of [5, 5, 1]-code) (see above)
- construction XX applied to C1 = C([1022,24]), C2 = C([0,25]), C3 = C1 + C2 = C([0,24]), and C∩ = C1 ∩ C2 = C([1022,25]) [i] based on
- 153 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 16 times 0, 1, 27 times 0, 1, 40 times 0, 1, 51 times 0) [i] based on linear OA(4101, 1033, F4, 27) (dual of [1033, 932, 28]-code), using
(109−27, 109, 189666)-Net in Base 4 — Upper bound on s
There is no (82, 109, 189667)-net in base 4, because
- 1 times m-reduction [i] would yield (82, 108, 189667)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 105316 555631 694380 734933 956574 909771 068912 041536 772991 102634 889200 > 4108 [i]