Best Known (79, 79+15, s)-Nets in Base 2
(79, 79+15, 1172)-Net over F2 — Constructive and digital
Digital (79, 94, 1172)-net over F2, using
- 21 times duplication [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
(79, 79+15, 2052)-Net over F2 — Digital
Digital (79, 94, 2052)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(294, 2052, F2, 4, 15) (dual of [(2052, 4), 8114, 16]-NRT-code), using
- OOA 4-folding [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
- OOA 4-folding [i] based on linear OA(294, 8208, F2, 15) (dual of [8208, 8114, 16]-code), using
(79, 79+15, 33743)-Net in Base 2 — Upper bound on s
There is no (79, 94, 33744)-net in base 2, because
- 1 times m-reduction [i] would yield (79, 93, 33744)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 9904 808164 039650 775236 362605 > 293 [i]