Best Known (26, 26+8, s)-Nets in Base 2
(26, 26+8, 78)-Net over F2 — Constructive and digital
Digital (26, 34, 78)-net over F2, using
- 22 times duplication [i] based on digital (24, 32, 78)-net over F2, using
- net defined by OOA [i] based on linear OOA(232, 78, F2, 8, 8) (dual of [(78, 8), 592, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(232, 78, F2, 7, 8) (dual of [(78, 7), 514, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(211, 47, F2, 7, 4) (dual of [(47, 7), 318, 5]-NRT-code), using
- appending 3 arbitrary columns [i] based on linear OOA(211, 47, F2, 4, 4) (dual of [(47, 4), 177, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(211, 47, F2, 3, 4) (dual of [(47, 3), 130, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (7, 11, 47)-net over F2, using
- appending kth column [i] based on linear OOA(211, 47, F2, 3, 4) (dual of [(47, 3), 130, 5]-NRT-code), using
- appending 3 arbitrary columns [i] based on linear OOA(211, 47, F2, 4, 4) (dual of [(47, 4), 177, 5]-NRT-code), using
- linear OOA(221, 39, F2, 7, 8) (dual of [(39, 7), 252, 9]-NRT-code), using
- extracting embedded OOA [i] based on digital (13, 21, 39)-net over F2, using
- linear OOA(211, 47, F2, 7, 4) (dual of [(47, 7), 318, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(232, 78, F2, 7, 8) (dual of [(78, 7), 514, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(232, 78, F2, 8, 8) (dual of [(78, 8), 592, 9]-NRT-code), using
(26, 26+8, 136)-Net over F2 — Digital
Digital (26, 34, 136)-net over F2, using
- net defined by OOA [i] based on linear OOA(234, 136, F2, 8, 8) (dual of [(136, 8), 1054, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(234, 136, F2, 7, 8) (dual of [(136, 7), 918, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(234, 136, F2, 2, 8) (dual of [(136, 2), 238, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(234, 272, F2, 8) (dual of [272, 238, 9]-code), using
- 1 times truncation [i] based on linear OA(235, 273, F2, 9) (dual of [273, 238, 10]-code), using
- construction XX applied to C1 = C([253,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([253,6]) [i] based on
- linear OA(225, 255, F2, 7) (dual of [255, 230, 8]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(225, 255, F2, 7) (dual of [255, 230, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(233, 255, F2, 9) (dual of [255, 222, 10]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(217, 255, F2, 5) (dual of [255, 238, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(21, 9, F2, 1) (dual of [9, 8, 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, 9, F2, 1) (dual of [9, 8, 2]-code) (see above)
- construction XX applied to C1 = C([253,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([253,6]) [i] based on
- 1 times truncation [i] based on linear OA(235, 273, F2, 9) (dual of [273, 238, 10]-code), using
- OOA 2-folding [i] based on linear OA(234, 272, F2, 8) (dual of [272, 238, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(234, 136, F2, 2, 8) (dual of [(136, 2), 238, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(234, 136, F2, 7, 8) (dual of [(136, 7), 918, 9]-NRT-code), using
(26, 26+8, 795)-Net in Base 2 — Upper bound on s
There is no (26, 34, 796)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 17193 678606 > 234 [i]