Best Known (158−15, 158, s)-Nets in Base 2
(158−15, 158, 599189)-Net over F2 — Constructive and digital
Digital (143, 158, 599189)-net over F2, using
- 22 times duplication [i] based on digital (141, 156, 599189)-net over F2, using
- net defined by OOA [i] based on linear OOA(2156, 599189, F2, 15, 15) (dual of [(599189, 15), 8987679, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2156, 4194324, F2, 15) (dual of [4194324, 4194168, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2156, 4194327, F2, 15) (dual of [4194327, 4194171, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2155, 4194304, F2, 15) (dual of [4194304, 4194149, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2133, 4194304, F2, 13) (dual of [4194304, 4194171, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(21, 23, F2, 1) (dual of [23, 22, 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 X applied to Ce(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(2156, 4194327, F2, 15) (dual of [4194327, 4194171, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2156, 4194324, F2, 15) (dual of [4194324, 4194168, 16]-code), using
- net defined by OOA [i] based on linear OOA(2156, 599189, F2, 15, 15) (dual of [(599189, 15), 8987679, 16]-NRT-code), using
(158−15, 158, 699055)-Net over F2 — Digital
Digital (143, 158, 699055)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2158, 699055, F2, 6, 15) (dual of [(699055, 6), 4194172, 16]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2158, 4194330, F2, 15) (dual of [4194330, 4194172, 16]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2156, 4194328, F2, 15) (dual of [4194328, 4194172, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2155, 4194304, F2, 15) (dual of [4194304, 4194149, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2133, 4194304, F2, 13) (dual of [4194304, 4194171, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(223, 24, F2, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,2)), using
- dual of repetition code with length 24 [i]
- linear OA(21, 24, F2, 1) (dual of [24, 23, 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(14) ⊂ Ce(12) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2156, 4194328, F2, 15) (dual of [4194328, 4194172, 16]-code), using
- OOA 6-folding [i] based on linear OA(2158, 4194330, F2, 15) (dual of [4194330, 4194172, 16]-code), using
(158−15, 158, large)-Net in Base 2 — Upper bound on s
There is no (143, 158, large)-net in base 2, because
- 13 times m-reduction [i] would yield (143, 145, large)-net in base 2, but