Best Known (172−17, 172, s)-Nets in Base 2
(172−17, 172, 262147)-Net over F2 — Constructive and digital
Digital (155, 172, 262147)-net over F2, using
- net defined by OOA [i] based on linear OOA(2172, 262147, F2, 17, 17) (dual of [(262147, 17), 4456327, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2172, 2097177, F2, 17) (dual of [2097177, 2097005, 18]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2170, 2097175, F2, 17) (dual of [2097175, 2097005, 18]-code), using
- construction X4 applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2169, 2097152, F2, 17) (dual of [2097152, 2096983, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2148, 2097152, F2, 15) (dual of [2097152, 2097004, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(222, 23, F2, 21) (dual of [23, 1, 22]-code), using
- strength reduction [i] based on linear OA(222, 23, F2, 22) (dual of [23, 1, 23]-code or 23-arc in PG(21,2)), using
- dual of repetition code with length 23 [i]
- strength reduction [i] based on linear OA(222, 23, F2, 22) (dual of [23, 1, 23]-code or 23-arc in PG(21,2)), using
- 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 X4 applied to Ce(16) ⊂ Ce(14) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2170, 2097175, F2, 17) (dual of [2097175, 2097005, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2172, 2097177, F2, 17) (dual of [2097177, 2097005, 18]-code), using
(172−17, 172, 349529)-Net over F2 — Digital
Digital (155, 172, 349529)-net over F2, using
- 22 times duplication [i] based on digital (153, 170, 349529)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2170, 349529, F2, 6, 17) (dual of [(349529, 6), 2097004, 18]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2170, 2097174, F2, 17) (dual of [2097174, 2097004, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2169, 2097152, F2, 17) (dual of [2097152, 2096983, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2148, 2097152, F2, 15) (dual of [2097152, 2097004, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 22, F2, 1) (dual of [22, 21, 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(16) ⊂ Ce(14) [i] based on
- OOA 6-folding [i] based on linear OA(2170, 2097174, F2, 17) (dual of [2097174, 2097004, 18]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2170, 349529, F2, 6, 17) (dual of [(349529, 6), 2097004, 18]-NRT-code), using
(172−17, 172, large)-Net in Base 2 — Upper bound on s
There is no (155, 172, large)-net in base 2, because
- 15 times m-reduction [i] would yield (155, 157, large)-net in base 2, but