Best Known (81, 81+14, s)-Nets in Base 2
(81, 81+14, 1172)-Net over F2 — Constructive and digital
Digital (81, 95, 1172)-net over F2, using
- 22 times duplication [i] based on digital (79, 93, 1172)-net over F2, using
- t-expansion [i] based on digital (78, 93, 1172)-net over F2, using
- net defined by OOA [i] based on linear OOA(293, 1172, F2, 15, 15) (dual of [(1172, 15), 17487, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(293, 8205, F2, 15) (dual of [8205, 8112, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(293, 8206, F2, 15) (dual of [8206, 8113, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(292, 8192, F2, 15) (dual of [8192, 8100, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(279, 8192, F2, 13) (dual of [8192, 8113, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(293, 8206, F2, 15) (dual of [8206, 8113, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(293, 8205, F2, 15) (dual of [8205, 8112, 16]-code), using
- net defined by OOA [i] based on linear OOA(293, 1172, F2, 15, 15) (dual of [(1172, 15), 17487, 16]-NRT-code), using
- t-expansion [i] based on digital (78, 93, 1172)-net over F2, using
(81, 81+14, 2650)-Net over F2 — Digital
Digital (81, 95, 2650)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(295, 2650, F2, 3, 14) (dual of [(2650, 3), 7855, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(295, 2736, F2, 3, 14) (dual of [(2736, 3), 8113, 15]-NRT-code), using
- 21 times duplication [i] based on linear OOA(294, 2736, F2, 3, 14) (dual of [(2736, 3), 8114, 15]-NRT-code), using
- OOA 3-folding [i] based on linear OA(294, 8208, F2, 14) (dual of [8208, 8114, 15]-code), using
- strength reduction [i] based on linear OA(294, 8208, F2, 15) (dual of [8208, 8114, 16]-code), using
- adding a parity check bit [i] based on linear OA(293, 8207, F2, 14) (dual of [8207, 8114, 15]-code), using
- construction X4 applied to C([0,14]) ⊂ C([1,12]) [i] based on
- linear OA(292, 8191, F2, 15) (dual of [8191, 8099, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(278, 8191, F2, 12) (dual of [8191, 8113, 13]-code), using the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(215, 16, F2, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,2)), using
- dual of repetition code with length 16 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 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 C([0,14]) ⊂ C([1,12]) [i] based on
- adding a parity check bit [i] based on linear OA(293, 8207, F2, 14) (dual of [8207, 8114, 15]-code), using
- strength reduction [i] based on linear OA(294, 8208, F2, 15) (dual of [8208, 8114, 16]-code), using
- OOA 3-folding [i] based on linear OA(294, 8208, F2, 14) (dual of [8208, 8114, 15]-code), using
- 21 times duplication [i] based on linear OOA(294, 2736, F2, 3, 14) (dual of [(2736, 3), 8114, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(295, 2736, F2, 3, 14) (dual of [(2736, 3), 8113, 15]-NRT-code), using
(81, 81+14, 41135)-Net in Base 2 — Upper bound on s
There is no (81, 95, 41136)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 39615 280634 136369 297633 594101 > 295 [i]