Best Known (82, 82+30, s)-Nets in Base 4
(82, 82+30, 531)-Net over F4 — Constructive and digital
Digital (82, 112, 531)-net over F4, using
- 41 times duplication [i] based on digital (81, 111, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 37, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 37, 177)-net over F64, using
(82, 82+30, 897)-Net over F4 — Digital
Digital (82, 112, 897)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4112, 897, F4, 30) (dual of [897, 785, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4112, 1034, F4, 30) (dual of [1034, 922, 31]-code), using
- construction XX applied to C1 = C([313,341]), C2 = C([315,342]), C3 = C1 + C2 = C([315,341]), and C∩ = C1 ∩ C2 = C([313,342]) [i] based on
- linear OA(4106, 1023, F4, 29) (dual of [1023, 917, 30]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {313,314,…,341}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4106, 1023, F4, 28) (dual of [1023, 917, 29]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {315,316,…,342}, and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4111, 1023, F4, 30) (dual of [1023, 912, 31]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {313,314,…,342}, and designed minimum distance d ≥ |I|+1 = 31 [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 = {315,316,…,341}, and designed minimum distance d ≥ |I|+1 = 28 [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([313,341]), C2 = C([315,342]), C3 = C1 + C2 = C([315,341]), and C∩ = C1 ∩ C2 = C([313,342]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4112, 1034, F4, 30) (dual of [1034, 922, 31]-code), using
(82, 82+30, 66980)-Net in Base 4 — Upper bound on s
There is no (82, 112, 66981)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 26 962522 628477 394869 963883 228018 604605 883681 808189 229245 889302 158048 > 4112 [i]