Best Known (38−9, 38, s)-Nets in Base 2
(38−9, 38, 130)-Net over F2 — Constructive and digital
Digital (29, 38, 130)-net over F2, using
- net defined by OOA [i] based on linear OOA(238, 130, F2, 9, 9) (dual of [(130, 9), 1132, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(238, 130, F2, 8, 9) (dual of [(130, 8), 1002, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(238, 521, F2, 9) (dual of [521, 483, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(238, 522, F2, 9) (dual of [522, 484, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(237, 512, F2, 9) (dual of [512, 475, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(228, 512, F2, 7) (dual of [512, 484, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(238, 522, F2, 9) (dual of [522, 484, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(238, 521, F2, 9) (dual of [521, 483, 10]-code), using
- appending kth column [i] based on linear OOA(238, 130, F2, 8, 9) (dual of [(130, 8), 1002, 10]-NRT-code), using
(38−9, 38, 184)-Net over F2 — Digital
Digital (29, 38, 184)-net over F2, using
- net defined by OOA [i] based on linear OOA(238, 184, F2, 9, 9) (dual of [(184, 9), 1618, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(238, 184, F2, 8, 9) (dual of [(184, 8), 1434, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(238, 184, F2, 2, 9) (dual of [(184, 2), 330, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(238, 261, F2, 2, 9) (dual of [(261, 2), 484, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(238, 522, F2, 9) (dual of [522, 484, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(237, 512, F2, 9) (dual of [512, 475, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(228, 512, F2, 7) (dual of [512, 484, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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(8) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(238, 522, F2, 9) (dual of [522, 484, 10]-code), using
- discarding factors / shortening the dual code based on linear OOA(238, 261, F2, 2, 9) (dual of [(261, 2), 484, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(238, 184, F2, 2, 9) (dual of [(184, 2), 330, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(238, 184, F2, 8, 9) (dual of [(184, 8), 1434, 10]-NRT-code), using
(38−9, 38, 1342)-Net in Base 2 — Upper bound on s
There is no (29, 38, 1343)-net in base 2, because
- 1 times m-reduction [i] would yield (29, 37, 1343)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 137778 203469 > 237 [i]