Best Known (79, 79+22, s)-Nets in Base 2
(79, 79+22, 152)-Net over F2 — Constructive and digital
Digital (79, 101, 152)-net over F2, using
- 21 times duplication [i] based on digital (78, 100, 152)-net over F2, using
- trace code for nets [i] based on digital (3, 25, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- trace code for nets [i] based on digital (3, 25, 38)-net over F16, using
(79, 79+22, 265)-Net over F2 — Digital
Digital (79, 101, 265)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2101, 265, F2, 2, 22) (dual of [(265, 2), 429, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2101, 530, F2, 22) (dual of [530, 429, 23]-code), using
- 1 times truncation [i] based on linear OA(2102, 531, F2, 23) (dual of [531, 429, 24]-code), using
- construction XX applied to C1 = C([509,18]), C2 = C([0,20]), C3 = C1 + C2 = C([0,18]), 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(291, 511, F2, 21) (dual of [511, 420, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [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(282, 511, F2, 19) (dual of [511, 429, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [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, 10, F2, 1) (dual of [10, 9, 2]-code) (see above)
- construction XX applied to C1 = C([509,18]), C2 = C([0,20]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([509,20]) [i] based on
- 1 times truncation [i] based on linear OA(2102, 531, F2, 23) (dual of [531, 429, 24]-code), using
- OOA 2-folding [i] based on linear OA(2101, 530, F2, 22) (dual of [530, 429, 23]-code), using
(79, 79+22, 2835)-Net in Base 2 — Upper bound on s
There is no (79, 101, 2836)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2 543441 037012 354586 844013 693026 > 2101 [i]