Best Known (24, 24+9, s)-Nets in Base 4
(24, 24+9, 258)-Net over F4 — Constructive and digital
Digital (24, 33, 258)-net over F4, using
- net defined by OOA [i] based on linear OOA(433, 258, F4, 9, 9) (dual of [(258, 9), 2289, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(433, 1033, F4, 9) (dual of [1033, 1000, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(433, 1035, F4, 9) (dual of [1035, 1002, 10]-code), using
- construction XX applied to C1 = C([339,345]), C2 = C([337,343]), C3 = C1 + C2 = C([339,343]), and C∩ = C1 ∩ C2 = C([337,345]) [i] based on
- linear OA(426, 1023, F4, 7) (dual of [1023, 997, 8]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {339,340,…,345}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(426, 1023, F4, 7) (dual of [1023, 997, 8]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {337,338,…,343}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(431, 1023, F4, 9) (dual of [1023, 992, 10]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {337,338,…,345}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(421, 1023, F4, 5) (dual of [1023, 1002, 6]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {339,340,341,342,343}, and designed minimum distance d ≥ |I|+1 = 6 [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(41, 6, F4, 1) (dual of [6, 5, 2]-code) (see above)
- construction XX applied to C1 = C([339,345]), C2 = C([337,343]), C3 = C1 + C2 = C([339,343]), and C∩ = C1 ∩ C2 = C([337,345]) [i] based on
- discarding factors / shortening the dual code based on linear OA(433, 1035, F4, 9) (dual of [1035, 1002, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(433, 1033, F4, 9) (dual of [1033, 1000, 10]-code), using
(24, 24+9, 387)-Net in Base 4 — Constructive
(24, 33, 387)-net in base 4, using
- trace code for nets [i] based on (2, 11, 129)-net in base 64, using
- 3 times m-reduction [i] based on (2, 14, 129)-net in base 64, using
- base change [i] based on digital (0, 12, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 12, 129)-net over F128, using
- 3 times m-reduction [i] based on (2, 14, 129)-net in base 64, using
(24, 24+9, 632)-Net over F4 — Digital
Digital (24, 33, 632)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(433, 632, F4, 9) (dual of [632, 599, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(433, 1032, F4, 9) (dual of [1032, 999, 10]-code), using
- construction XX applied to Ce(8) ⊂ Ce(6) ⊂ Ce(5) [i] based on
- linear OA(431, 1024, F4, 9) (dual of [1024, 993, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(426, 1024, F4, 7) (dual of [1024, 998, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(421, 1024, F4, 6) (dual of [1024, 1003, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 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, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(8) ⊂ Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(433, 1032, F4, 9) (dual of [1032, 999, 10]-code), using
(24, 24+9, 48348)-Net in Base 4 — Upper bound on s
There is no (24, 33, 48349)-net in base 4, because
- 1 times m-reduction [i] would yield (24, 32, 48349)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 18 446991 913724 022139 > 432 [i]