Best Known (42−10, 42, s)-Nets in Base 7
(42−10, 42, 3363)-Net over F7 — Constructive and digital
Digital (32, 42, 3363)-net over F7, using
- net defined by OOA [i] based on linear OOA(742, 3363, F7, 10, 10) (dual of [(3363, 10), 33588, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(742, 16815, F7, 10) (dual of [16815, 16773, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(742, 16818, F7, 10) (dual of [16818, 16776, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(741, 16807, F7, 10) (dual of [16807, 16766, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(742, 16818, F7, 10) (dual of [16818, 16776, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(742, 16815, F7, 10) (dual of [16815, 16773, 11]-code), using
(42−10, 42, 13444)-Net over F7 — Digital
Digital (32, 42, 13444)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(742, 13444, F7, 10) (dual of [13444, 13402, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(742, 16818, F7, 10) (dual of [16818, 16776, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(741, 16807, F7, 10) (dual of [16807, 16766, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(742, 16818, F7, 10) (dual of [16818, 16776, 11]-code), using
(42−10, 42, 5451403)-Net in Base 7 — Upper bound on s
There is no (32, 42, 5451404)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 311973 695193 919816 416721 907107 418857 > 742 [i]