Best Known (115−30, 115, s)-Nets in Base 4
(115−30, 115, 531)-Net over F4 — Constructive and digital
Digital (85, 115, 531)-net over F4, using
- 2 times m-reduction [i] based on digital (85, 117, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 39, 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, 39, 177)-net over F64, using
(115−30, 115, 1043)-Net over F4 — Digital
Digital (85, 115, 1043)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4115, 1043, F4, 30) (dual of [1043, 928, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4115, 1045, F4, 30) (dual of [1045, 930, 31]-code), using
- construction XX applied to C1 = C([1019,24]), C2 = C([0,25]), C3 = C1 + C2 = C([0,24]), and C∩ = C1 ∩ C2 = C([1019,25]) [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 = {−4,−3,…,24}, and designed minimum distance d ≥ |I|+1 = 30 [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(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 = {−4,−3,…,25}, and designed minimum distance d ≥ |I|+1 = 31 [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(44, 17, F4, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), 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([1019,24]), C2 = C([0,25]), C3 = C1 + C2 = C([0,24]), and C∩ = C1 ∩ C2 = C([1019,25]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4115, 1045, F4, 30) (dual of [1045, 930, 31]-code), using
(115−30, 115, 88385)-Net in Base 4 — Upper bound on s
There is no (85, 115, 88386)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1725 662062 390310 878014 855043 376697 078692 928509 863414 896856 117083 840160 > 4115 [i]