Best Known (3, 7, s)-Nets in Base 128
(3, 7, 8193)-Net over F128 — Constructive and digital
Digital (3, 7, 8193)-net over F128, using
- net defined by OOA [i] based on linear OOA(1287, 8193, F128, 4, 4) (dual of [(8193, 4), 32765, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(1287, 16386, F128, 4) (dual of [16386, 16379, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(1287, 16384, F128, 4) (dual of [16384, 16377, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1285, 16384, F128, 3) (dual of [16384, 16379, 4]-code or 16384-cap in PG(4,128)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(1287, 16386, F128, 4) (dual of [16386, 16379, 5]-code), using
(3, 7, 16386)-Net over F128 — Digital
Digital (3, 7, 16386)-net over F128, using
- net defined by OOA [i] based on linear OOA(1287, 16386, F128, 4, 4) (dual of [(16386, 4), 65537, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(1287, 16386, F128, 3, 4) (dual of [(16386, 3), 49151, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1287, 16386, F128, 4) (dual of [16386, 16379, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(1287, 16384, F128, 4) (dual of [16384, 16377, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1285, 16384, F128, 3) (dual of [16384, 16379, 4]-code or 16384-cap in PG(4,128)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1287, 16386, F128, 4) (dual of [16386, 16379, 5]-code), using
- appending kth column [i] based on linear OOA(1287, 16386, F128, 3, 4) (dual of [(16386, 3), 49151, 5]-NRT-code), using
(3, 7, 32640)-Net in Base 128 — Constructive
(3, 7, 32640)-net in base 128, using
- 1 times m-reduction [i] based on (3, 8, 32640)-net in base 128, using
- base change [i] based on digital (2, 7, 32640)-net over F256, using
- net defined by OOA [i] based on linear OOA(2567, 32640, F256, 5, 5) (dual of [(32640, 5), 163193, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2567, 65281, F256, 5) (dual of [65281, 65274, 6]-code), using
- net defined by OOA [i] based on linear OOA(2567, 32640, F256, 5, 5) (dual of [(32640, 5), 163193, 6]-NRT-code), using
- base change [i] based on digital (2, 7, 32640)-net over F256, using
(3, 7, 264207)-Net in Base 128 — Upper bound on s
There is no (3, 7, 264208)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 562 951614 364777 > 1287 [i]