Best Known (128−13, 128, s)-Nets in Base 2
(128−13, 128, 349529)-Net over F2 — Constructive and digital
Digital (115, 128, 349529)-net over F2, using
- net defined by OOA [i] based on linear OOA(2128, 349529, F2, 13, 13) (dual of [(349529, 13), 4543749, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2128, 2097175, F2, 13) (dual of [2097175, 2097047, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(2127, 2097152, F2, 13) (dual of [2097152, 2097025, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2106, 2097152, F2, 11) (dual of [2097152, 2097046, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(12) ⊂ Ce(10) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(2128, 2097175, F2, 13) (dual of [2097175, 2097047, 14]-code), using
(128−13, 128, 419435)-Net over F2 — Digital
Digital (115, 128, 419435)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2128, 419435, F2, 5, 13) (dual of [(419435, 5), 2097047, 14]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2128, 2097175, F2, 13) (dual of [2097175, 2097047, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(2127, 2097152, F2, 13) (dual of [2097152, 2097025, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2106, 2097152, F2, 11) (dual of [2097152, 2097046, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(12) ⊂ Ce(10) [i] based on
- OOA 5-folding [i] based on linear OA(2128, 2097175, F2, 13) (dual of [2097175, 2097047, 14]-code), using
(128−13, 128, 7047306)-Net in Base 2 — Upper bound on s
There is no (115, 128, 7047307)-net in base 2, because
- 1 times m-reduction [i] would yield (115, 127, 7047307)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 170 141317 814496 171913 728705 352365 919594 > 2127 [i]