Best Known (122−32, 122, s)-Nets in Base 4
(122−32, 122, 531)-Net over F4 — Constructive and digital
Digital (90, 122, 531)-net over F4, using
- t-expansion [i] based on digital (89, 122, 531)-net over F4, using
- 1 times m-reduction [i] based on digital (89, 123, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 41, 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, 41, 177)-net over F64, using
- 1 times m-reduction [i] based on digital (89, 123, 531)-net over F4, using
(122−32, 122, 1039)-Net over F4 — Digital
Digital (90, 122, 1039)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4122, 1039, F4, 32) (dual of [1039, 917, 33]-code), using
- construction XX applied to C1 = C([1021,28]), C2 = C([0,29]), C3 = C1 + C2 = C([0,28]), and C∩ = C1 ∩ C2 = C([1021,29]) [i] based on
- linear OA(4116, 1023, F4, 31) (dual of [1023, 907, 32]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−2,−1,…,28}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(4111, 1023, F4, 30) (dual of [1023, 912, 31]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,29], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4121, 1023, F4, 32) (dual of [1023, 902, 33]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−2,−1,…,29}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4106, 1023, F4, 29) (dual of [1023, 917, 30]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 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([1021,28]), C2 = C([0,29]), C3 = C1 + C2 = C([0,28]), and C∩ = C1 ∩ C2 = C([1021,29]) [i] based on
(122−32, 122, 88320)-Net in Base 4 — Upper bound on s
There is no (90, 122, 88321)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 28 271813 317973 845529 696594 986825 651304 292749 757638 427194 413920 914829 944944 > 4122 [i]