Best Known (30, 30+7, s)-Nets in Base 5
(30, 30+7, 26044)-Net over F5 — Constructive and digital
Digital (30, 37, 26044)-net over F5, using
- net defined by OOA [i] based on linear OOA(537, 26044, F5, 7, 7) (dual of [(26044, 7), 182271, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(537, 78133, F5, 7) (dual of [78133, 78096, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(537, 78134, F5, 7) (dual of [78134, 78097, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(536, 78125, F5, 7) (dual of [78125, 78089, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(529, 78125, F5, 6) (dual of [78125, 78096, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(58, 9, F5, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,5)), using
- dual of repetition code with length 9 [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(537, 78134, F5, 7) (dual of [78134, 78097, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(537, 78133, F5, 7) (dual of [78133, 78096, 8]-code), using
(30, 30+7, 70201)-Net over F5 — Digital
Digital (30, 37, 70201)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(537, 70201, F5, 7) (dual of [70201, 70164, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(537, 78134, F5, 7) (dual of [78134, 78097, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(536, 78125, F5, 7) (dual of [78125, 78089, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(529, 78125, F5, 6) (dual of [78125, 78096, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(58, 9, F5, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,5)), using
- dual of repetition code with length 9 [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(537, 78134, F5, 7) (dual of [78134, 78097, 8]-code), using
(30, 30+7, large)-Net in Base 5 — Upper bound on s
There is no (30, 37, large)-net in base 5, because
- 5 times m-reduction [i] would yield (30, 32, large)-net in base 5, but