Best Known (244−19, 244, s)-Nets in Base 2
(244−19, 244, 932138)-Net over F2 — Constructive and digital
Digital (225, 244, 932138)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (27, 36, 72)-net over F2, using
- trace code for nets [i] based on digital (3, 12, 24)-net over F8, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- the Klein quartic over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- trace code for nets [i] based on digital (3, 12, 24)-net over F8, using
- digital (189, 208, 932066)-net over F2, using
- net defined by OOA [i] based on linear OOA(2208, 932066, F2, 19, 19) (dual of [(932066, 19), 17709046, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2208, 8388595, F2, 19) (dual of [8388595, 8388387, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2208, 8388595, F2, 19) (dual of [8388595, 8388387, 20]-code), using
- net defined by OOA [i] based on linear OOA(2208, 932066, F2, 19, 19) (dual of [(932066, 19), 17709046, 20]-NRT-code), using
- digital (27, 36, 72)-net over F2, using
(244−19, 244, 1677857)-Net over F2 — Digital
Digital (225, 244, 1677857)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2244, 1677857, F2, 5, 19) (dual of [(1677857, 5), 8389041, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(236, 137, F2, 5, 9) (dual of [(137, 5), 649, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(236, 137, F2, 2, 9) (dual of [(137, 2), 238, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(236, 274, F2, 9) (dual of [274, 238, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(236, 275, F2, 9) (dual of [275, 239, 10]-code), using
- adding a parity check bit [i] based on linear OA(235, 274, F2, 8) (dual of [274, 239, 9]-code), using
- construction XX applied to C1 = C([253,4]), C2 = C([1,6]), C3 = C1 + C2 = C([1,4]), and C∩ = C1 ∩ C2 = C([253,6]) [i] based on
- linear OA(225, 255, F2, 7) (dual of [255, 230, 8]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(224, 255, F2, 6) (dual of [255, 231, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(233, 255, F2, 9) (dual of [255, 222, 10]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(216, 255, F2, 4) (dual of [255, 239, 5]-code), using the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(21, 10, F2, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(21, 9, F2, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s (see above)
- construction XX applied to C1 = C([253,4]), C2 = C([1,6]), C3 = C1 + C2 = C([1,4]), and C∩ = C1 ∩ C2 = C([253,6]) [i] based on
- adding a parity check bit [i] based on linear OA(235, 274, F2, 8) (dual of [274, 239, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(236, 275, F2, 9) (dual of [275, 239, 10]-code), using
- OOA 2-folding [i] based on linear OA(236, 274, F2, 9) (dual of [274, 238, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(236, 137, F2, 2, 9) (dual of [(137, 2), 238, 10]-NRT-code), using
- linear OOA(2208, 1677720, F2, 5, 19) (dual of [(1677720, 5), 8388392, 20]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2208, 8388600, F2, 19) (dual of [8388600, 8388392, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- OOA 5-folding [i] based on linear OA(2208, 8388600, F2, 19) (dual of [8388600, 8388392, 20]-code), using
- linear OOA(236, 137, F2, 5, 9) (dual of [(137, 5), 649, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(244−19, 244, large)-Net in Base 2 — Upper bound on s
There is no (225, 244, large)-net in base 2, because
- 17 times m-reduction [i] would yield (225, 227, large)-net in base 2, but