Best Known (253−19, 253, s)-Nets in Base 4
(253−19, 253, 3728522)-Net over F4 — Constructive and digital
Digital (234, 253, 3728522)-net over F4, using
- (u, u+v)-construction [i] based on
- 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
- net defined by OOA [i] based on linear OOA(433, 258, F4, 9, 9) (dual of [(258, 9), 2289, 10]-NRT-code), using
- digital (201, 220, 3728264)-net over F4, using
- trace code for nets [i] based on digital (36, 55, 932066)-net over F256, using
- net defined by OOA [i] based on linear OOA(25655, 932066, F256, 19, 19) (dual of [(932066, 19), 17709199, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(25655, 8388595, F256, 19) (dual of [8388595, 8388540, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(25655, large, F256, 19) (dual of [large, large−55, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(25655, large, F256, 19) (dual of [large, large−55, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(25655, 8388595, F256, 19) (dual of [8388595, 8388540, 20]-code), using
- net defined by OOA [i] based on linear OOA(25655, 932066, F256, 19, 19) (dual of [(932066, 19), 17709199, 20]-NRT-code), using
- trace code for nets [i] based on digital (36, 55, 932066)-net over F256, using
- digital (24, 33, 258)-net over F4, using
(253−19, 253, 3728651)-Net in Base 4 — Constructive
(234, 253, 3728651)-net in base 4, using
- (u, u+v)-construction [i] based on
- (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
- trace code for nets [i] based on (2, 11, 129)-net in base 64, using
- digital (201, 220, 3728264)-net over F4, using
- trace code for nets [i] based on digital (36, 55, 932066)-net over F256, using
- net defined by OOA [i] based on linear OOA(25655, 932066, F256, 19, 19) (dual of [(932066, 19), 17709199, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(25655, 8388595, F256, 19) (dual of [8388595, 8388540, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(25655, large, F256, 19) (dual of [large, large−55, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(25655, large, F256, 19) (dual of [large, large−55, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(25655, 8388595, F256, 19) (dual of [8388595, 8388540, 20]-code), using
- net defined by OOA [i] based on linear OOA(25655, 932066, F256, 19, 19) (dual of [(932066, 19), 17709199, 20]-NRT-code), using
- trace code for nets [i] based on digital (36, 55, 932066)-net over F256, using
- (24, 33, 387)-net in base 4, using
(253−19, 253, large)-Net over F4 — Digital
Digital (234, 253, large)-net over F4, using
- t-expansion [i] based on digital (231, 253, large)-net over F4, using
- 4 times m-reduction [i] based on digital (231, 257, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4257, large, F4, 26) (dual of [large, large−257, 27]-code), using
- 28 times code embedding in larger space [i] based on linear OA(4229, large, F4, 26) (dual of [large, large−229, 27]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [i]
- 28 times code embedding in larger space [i] based on linear OA(4229, large, F4, 26) (dual of [large, large−229, 27]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4257, large, F4, 26) (dual of [large, large−257, 27]-code), using
- 4 times m-reduction [i] based on digital (231, 257, large)-net over F4, using
(253−19, 253, large)-Net in Base 4 — Upper bound on s
There is no (234, 253, large)-net in base 4, because
- 17 times m-reduction [i] would yield (234, 236, large)-net in base 4, but