Best Known (124, 124+29, s)-Nets in Base 2
(124, 124+29, 260)-Net over F2 — Constructive and digital
Digital (124, 153, 260)-net over F2, using
- t-expansion [i] based on digital (123, 153, 260)-net over F2, using
- 3 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
- 3 times m-reduction [i] based on digital (123, 156, 260)-net over F2, using
(124, 124+29, 537)-Net over F2 — Digital
Digital (124, 153, 537)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2153, 537, F2, 2, 29) (dual of [(537, 2), 921, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2153, 1074, F2, 29) (dual of [1074, 921, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2153, 1075, F2, 29) (dual of [1075, 922, 30]-code), using
- construction XX applied to C1 = C([1019,20]), C2 = C([0,24]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([1019,24]) [i] based on
- linear OA(2121, 1023, F2, 25) (dual of [1023, 902, 26]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,20}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2121, 1023, F2, 25) (dual of [1023, 902, 26]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,24], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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 = {−4,−3,…,24}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(2101, 1023, F2, 21) (dual of [1023, 922, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(26, 26, F2, 3) (dual of [26, 20, 4]-code or 26-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- linear OA(26, 26, F2, 3) (dual of [26, 20, 4]-code or 26-cap in PG(5,2)) (see above)
- construction XX applied to C1 = C([1019,20]), C2 = C([0,24]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([1019,24]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2153, 1075, F2, 29) (dual of [1075, 922, 30]-code), using
- OOA 2-folding [i] based on linear OA(2153, 1074, F2, 29) (dual of [1074, 921, 30]-code), using
(124, 124+29, 11194)-Net in Base 2 — Upper bound on s
There is no (124, 153, 11195)-net in base 2, because
- 1 times m-reduction [i] would yield (124, 152, 11195)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 5715 357791 269759 856534 485564 120057 169907 811164 > 2152 [i]