Best Known (77−11, 77, s)-Nets in Base 2
(77−11, 77, 6556)-Net over F2 — Constructive and digital
Digital (66, 77, 6556)-net over F2, using
- net defined by OOA [i] based on linear OOA(277, 6556, F2, 11, 11) (dual of [(6556, 11), 72039, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(277, 32781, F2, 11) (dual of [32781, 32704, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(277, 32784, F2, 11) (dual of [32784, 32707, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(276, 32768, F2, 11) (dual of [32768, 32692, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(261, 32768, F2, 9) (dual of [32768, 32707, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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 X applied to Ce(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(277, 32784, F2, 11) (dual of [32784, 32707, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(277, 32781, F2, 11) (dual of [32781, 32704, 12]-code), using
(77−11, 77, 8196)-Net over F2 — Digital
Digital (66, 77, 8196)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(277, 8196, F2, 4, 11) (dual of [(8196, 4), 32707, 12]-NRT-code), using
- OOA 4-folding [i] based on linear OA(277, 32784, F2, 11) (dual of [32784, 32707, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(276, 32768, F2, 11) (dual of [32768, 32692, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(261, 32768, F2, 9) (dual of [32768, 32707, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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 X applied to Ce(10) ⊂ Ce(8) [i] based on
- OOA 4-folding [i] based on linear OA(277, 32784, F2, 11) (dual of [32784, 32707, 12]-code), using
(77−11, 77, 98053)-Net in Base 2 — Upper bound on s
There is no (66, 77, 98054)-net in base 2, because
- 1 times m-reduction [i] would yield (66, 76, 98054)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 75561 459511 945247 349587 > 276 [i]