Best Known (80, 80+23, s)-Nets in Base 2
(80, 80+23, 144)-Net over F2 — Constructive and digital
Digital (80, 103, 144)-net over F2, using
- 21 times duplication [i] based on digital (79, 102, 144)-net over F2, using
- trace code for nets [i] based on digital (11, 34, 48)-net over F8, using
- net from sequence [i] based on digital (11, 47)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 11 and N(F) ≥ 48, using
- net from sequence [i] based on digital (11, 47)-sequence over F8, using
- trace code for nets [i] based on digital (11, 34, 48)-net over F8, using
(80, 80+23, 248)-Net over F2 — Digital
Digital (80, 103, 248)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2103, 248, F2, 2, 23) (dual of [(248, 2), 393, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2103, 266, F2, 2, 23) (dual of [(266, 2), 429, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2103, 532, F2, 23) (dual of [532, 429, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2103, 533, F2, 23) (dual of [533, 430, 24]-code), using
- adding a parity check bit [i] based on linear OA(2102, 532, F2, 22) (dual of [532, 430, 23]-code), using
- construction XX applied to C1 = C([509,18]), C2 = C([1,20]), C3 = C1 + C2 = C([1,18]), and C∩ = C1 ∩ C2 = C([509,20]) [i] based on
- linear OA(291, 511, F2, 21) (dual of [511, 420, 22]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,18}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(290, 511, F2, 20) (dual of [511, 421, 21]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2100, 511, F2, 23) (dual of [511, 411, 24]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,20}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(281, 511, F2, 18) (dual of [511, 430, 19]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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, using
- 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 (see above)
- construction XX applied to C1 = C([509,18]), C2 = C([1,20]), C3 = C1 + C2 = C([1,18]), and C∩ = C1 ∩ C2 = C([509,20]) [i] based on
- adding a parity check bit [i] based on linear OA(2102, 532, F2, 22) (dual of [532, 430, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2103, 533, F2, 23) (dual of [533, 430, 24]-code), using
- OOA 2-folding [i] based on linear OA(2103, 532, F2, 23) (dual of [532, 429, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(2103, 266, F2, 2, 23) (dual of [(266, 2), 429, 24]-NRT-code), using
(80, 80+23, 3020)-Net in Base 2 — Upper bound on s
There is no (80, 103, 3021)-net in base 2, because
- 1 times m-reduction [i] would yield (80, 102, 3021)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 5 077823 437693 208850 834289 376032 > 2102 [i]