Best Known (27, 27+11, s)-Nets in Base 7
(27, 27+11, 481)-Net over F7 — Constructive and digital
Digital (27, 38, 481)-net over F7, using
- net defined by OOA [i] based on linear OOA(738, 481, F7, 11, 11) (dual of [(481, 11), 5253, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(738, 2406, F7, 11) (dual of [2406, 2368, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(738, 2410, F7, 11) (dual of [2410, 2372, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(737, 2401, F7, 11) (dual of [2401, 2364, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(729, 2401, F7, 9) (dual of [2401, 2372, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(71, 9, F7, 1) (dual of [9, 8, 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(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(738, 2410, F7, 11) (dual of [2410, 2372, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(738, 2406, F7, 11) (dual of [2406, 2368, 12]-code), using
(27, 27+11, 2055)-Net over F7 — Digital
Digital (27, 38, 2055)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(738, 2055, F7, 11) (dual of [2055, 2017, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(738, 2410, F7, 11) (dual of [2410, 2372, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(737, 2401, F7, 11) (dual of [2401, 2364, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(729, 2401, F7, 9) (dual of [2401, 2372, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(71, 9, F7, 1) (dual of [9, 8, 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(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(738, 2410, F7, 11) (dual of [2410, 2372, 12]-code), using
(27, 27+11, 778769)-Net in Base 7 — Upper bound on s
There is no (27, 38, 778770)-net in base 7, because
- 1 times m-reduction [i] would yield (27, 37, 778770)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 18 562179 663774 357874 385729 093605 > 737 [i]