Best Known (91, 91+23, s)-Nets in Base 2
(91, 91+23, 196)-Net over F2 — Constructive and digital
Digital (91, 114, 196)-net over F2, using
- 22 times duplication [i] based on digital (89, 112, 196)-net over F2, using
- trace code for nets [i] based on digital (5, 28, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- trace code for nets [i] based on digital (5, 28, 49)-net over F16, using
(91, 91+23, 375)-Net over F2 — Digital
Digital (91, 114, 375)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2114, 375, F2, 2, 23) (dual of [(375, 2), 636, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2114, 523, F2, 2, 23) (dual of [(523, 2), 932, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2114, 1046, F2, 23) (dual of [1046, 932, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2114, 1047, F2, 23) (dual of [1047, 933, 24]-code), using
- adding a parity check bit [i] based on linear OA(2113, 1046, F2, 22) (dual of [1046, 933, 23]-code), using
- construction XX applied to C1 = C([1021,18]), C2 = C([1,20]), C3 = C1 + C2 = C([1,18]), and C∩ = C1 ∩ C2 = C([1021,20]) [i] based on
- linear OA(2101, 1023, F2, 21) (dual of [1023, 922, 22]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,18}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2100, 1023, F2, 20) (dual of [1023, 923, 21]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2111, 1023, F2, 23) (dual of [1023, 912, 24]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,20}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(290, 1023, F2, 18) (dual of [1023, 933, 19]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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([1021,18]), C2 = C([1,20]), C3 = C1 + C2 = C([1,18]), and C∩ = C1 ∩ C2 = C([1021,20]) [i] based on
- adding a parity check bit [i] based on linear OA(2113, 1046, F2, 22) (dual of [1046, 933, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2114, 1047, F2, 23) (dual of [1047, 933, 24]-code), using
- OOA 2-folding [i] based on linear OA(2114, 1046, F2, 23) (dual of [1046, 932, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(2114, 523, F2, 2, 23) (dual of [(523, 2), 932, 24]-NRT-code), using
(91, 91+23, 6057)-Net in Base 2 — Upper bound on s
There is no (91, 114, 6058)-net in base 2, because
- 1 times m-reduction [i] would yield (91, 113, 6058)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 10400 448678 759095 184033 549127 351332 > 2113 [i]