Best Known (123, 123+31, s)-Nets in Base 2
(123, 123+31, 260)-Net over F2 — Constructive and digital
Digital (123, 154, 260)-net over F2, using
- 2 times m-reduction [i] based on digital (123, 156, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 39, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 39, 65)-net over F16, using
(123, 123+31, 448)-Net over F2 — Digital
Digital (123, 154, 448)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2154, 448, F2, 2, 31) (dual of [(448, 2), 742, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2154, 523, F2, 2, 31) (dual of [(523, 2), 892, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2154, 1046, F2, 31) (dual of [1046, 892, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2154, 1047, F2, 31) (dual of [1047, 893, 32]-code), using
- adding a parity check bit [i] based on linear OA(2153, 1046, F2, 30) (dual of [1046, 893, 31]-code), using
- construction XX applied to C1 = C([1021,26]), C2 = C([1,28]), C3 = C1 + C2 = C([1,26]), and C∩ = C1 ∩ C2 = C([1021,28]) [i] based on
- linear OA(2141, 1023, F2, 29) (dual of [1023, 882, 30]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,26}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(2140, 1023, F2, 28) (dual of [1023, 883, 29]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2151, 1023, F2, 31) (dual of [1023, 872, 32]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,28}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2130, 1023, F2, 26) (dual of [1023, 893, 27]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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, 11, F2, 1) (dual of [11, 10, 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([1021,26]), C2 = C([1,28]), C3 = C1 + C2 = C([1,26]), and C∩ = C1 ∩ C2 = C([1021,28]) [i] based on
- adding a parity check bit [i] based on linear OA(2153, 1046, F2, 30) (dual of [1046, 893, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2154, 1047, F2, 31) (dual of [1047, 893, 32]-code), using
- OOA 2-folding [i] based on linear OA(2154, 1046, F2, 31) (dual of [1046, 892, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(2154, 523, F2, 2, 31) (dual of [(523, 2), 892, 32]-NRT-code), using
(123, 123+31, 7533)-Net in Base 2 — Upper bound on s
There is no (123, 154, 7534)-net in base 2, because
- 1 times m-reduction [i] would yield (123, 153, 7534)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 11424 958791 967694 049833 618853 884080 949002 017500 > 2153 [i]