Best Known (111−8, 111, s)-Nets in Base 2
(111−8, 111, 2097663)-Net over F2 — Constructive and digital
Digital (103, 111, 2097663)-net over F2, using
- net defined by OOA [i] based on linear OOA(2111, 2097663, F2, 8, 8) (dual of [(2097663, 8), 16781193, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2111, 2097663, F2, 7, 8) (dual of [(2097663, 7), 14683530, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(219, 513, F2, 7, 4) (dual of [(513, 7), 3572, 5]-NRT-code), using
- appending 3 arbitrary columns [i] based on linear OOA(219, 513, F2, 4, 4) (dual of [(513, 4), 2033, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(219, 513, F2, 3, 4) (dual of [(513, 3), 1520, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (15, 19, 513)-net over F2, using
- appending kth column [i] based on linear OOA(219, 513, F2, 3, 4) (dual of [(513, 3), 1520, 5]-NRT-code), using
- appending 3 arbitrary columns [i] based on linear OOA(219, 513, F2, 4, 4) (dual of [(513, 4), 2033, 5]-NRT-code), using
- linear OOA(292, 2097150, F2, 7, 8) (dual of [(2097150, 7), 14679958, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(292, 8388600, F2, 8) (dual of [8388600, 8388508, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(292, large, F2, 8) (dual of [large, large−92, 9]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(292, large, F2, 8) (dual of [large, large−92, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(292, 8388600, F2, 8) (dual of [8388600, 8388508, 9]-code), using
- linear OOA(219, 513, F2, 7, 4) (dual of [(513, 7), 3572, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(2111, 2097663, F2, 7, 8) (dual of [(2097663, 7), 14683530, 9]-NRT-code), using
(111−8, 111, 4194823)-Net over F2 — Digital
Digital (103, 111, 4194823)-net over F2, using
- net defined by OOA [i] based on linear OOA(2111, 4194823, F2, 8, 8) (dual of [(4194823, 8), 33558473, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2111, 4194823, F2, 7, 8) (dual of [(4194823, 7), 29363650, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2111, 4194823, F2, 2, 8) (dual of [(4194823, 2), 8389535, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(219, 522, F2, 2, 4) (dual of [(522, 2), 1025, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(219, 522, F2, 4) (dual of [522, 503, 5]-code), using
- 1 times truncation [i] based on linear OA(220, 523, F2, 5) (dual of [523, 503, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(219, 512, F2, 5) (dual of [512, 493, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(210, 512, F2, 3) (dual of [512, 502, 4]-code or 512-cap in PG(9,2)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(210, 11, F2, 9) (dual of [11, 1, 10]-code), using
- strength reduction [i] based on linear OA(210, 11, F2, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,2)), using
- dual of repetition code with length 11 [i]
- strength reduction [i] based on linear OA(210, 11, F2, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,2)), 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, using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- 1 times truncation [i] based on linear OA(220, 523, F2, 5) (dual of [523, 503, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(219, 522, F2, 4) (dual of [522, 503, 5]-code), using
- linear OOA(292, 4194301, F2, 2, 8) (dual of [(4194301, 2), 8388510, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(292, 8388602, F2, 8) (dual of [8388602, 8388510, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(292, large, F2, 8) (dual of [large, large−92, 9]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(292, large, F2, 8) (dual of [large, large−92, 9]-code), using
- OOA 2-folding [i] based on linear OA(292, 8388602, F2, 8) (dual of [8388602, 8388510, 9]-code), using
- linear OOA(219, 522, F2, 2, 4) (dual of [(522, 2), 1025, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2111, 4194823, F2, 2, 8) (dual of [(4194823, 2), 8389535, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2111, 4194823, F2, 7, 8) (dual of [(4194823, 7), 29363650, 9]-NRT-code), using
(111−8, 111, large)-Net in Base 2 — Upper bound on s
There is no (103, 111, large)-net in base 2, because
- 6 times m-reduction [i] would yield (103, 105, large)-net in base 2, but