Best Known (72−13, 72, s)-Nets in Base 2
(72−13, 72, 344)-Net over F2 — Constructive and digital
Digital (59, 72, 344)-net over F2, using
- net defined by OOA [i] based on linear OOA(272, 344, F2, 13, 13) (dual of [(344, 13), 4400, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(272, 2065, F2, 13) (dual of [2065, 1993, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(267, 2049, F2, 13) (dual of [2049, 1982, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(245, 2049, F2, 9) (dual of [2049, 2004, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(25, 16, F2, 3) (dual of [16, 11, 4]-code or 16-cap in PG(4,2)), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(272, 2065, F2, 13) (dual of [2065, 1993, 14]-code), using
(72−13, 72, 688)-Net over F2 — Digital
Digital (59, 72, 688)-net over F2, using
- 21 times duplication [i] based on digital (58, 71, 688)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(271, 688, F2, 3, 13) (dual of [(688, 3), 1993, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(271, 2064, F2, 13) (dual of [2064, 1993, 14]-code), using
- 2 times code embedding in larger space [i] based on linear OA(269, 2062, F2, 13) (dual of [2062, 1993, 14]-code), using
- adding a parity check bit [i] based on linear OA(268, 2061, F2, 12) (dual of [2061, 1993, 13]-code), using
- construction X4 applied to C([0,12]) ⊂ C([1,10]) [i] based on
- linear OA(267, 2047, F2, 13) (dual of [2047, 1980, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(255, 2047, F2, 10) (dual of [2047, 1992, 11]-code), using the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(213, 14, F2, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,2)), using
- dual of repetition code with length 14 [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 X4 applied to C([0,12]) ⊂ C([1,10]) [i] based on
- adding a parity check bit [i] based on linear OA(268, 2061, F2, 12) (dual of [2061, 1993, 13]-code), using
- 2 times code embedding in larger space [i] based on linear OA(269, 2062, F2, 13) (dual of [2062, 1993, 14]-code), using
- OOA 3-folding [i] based on linear OA(271, 2064, F2, 13) (dual of [2064, 1993, 14]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(271, 688, F2, 3, 13) (dual of [(688, 3), 1993, 14]-NRT-code), using
(72−13, 72, 10916)-Net in Base 2 — Upper bound on s
There is no (59, 72, 10917)-net in base 2, because
- 1 times m-reduction [i] would yield (59, 71, 10917)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 2362 189557 323355 033100 > 271 [i]