Best Known (30, 36, s)-Nets in Base 8
(30, 36, 699053)-Net over F8 — Constructive and digital
Digital (30, 36, 699053)-net over F8, using
- net defined by OOA [i] based on linear OOA(836, 699053, F8, 6, 6) (dual of [(699053, 6), 4194282, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(836, 2097159, F8, 6) (dual of [2097159, 2097123, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(836, 2097152, F8, 6) (dual of [2097152, 2097116, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(829, 2097152, F8, 5) (dual of [2097152, 2097123, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(80, 7, F8, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- OA 3-folding and stacking [i] based on linear OA(836, 2097159, F8, 6) (dual of [2097159, 2097123, 7]-code), using
(30, 36, 2097159)-Net over F8 — Digital
Digital (30, 36, 2097159)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(836, 2097159, F8, 6) (dual of [2097159, 2097123, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(836, 2097152, F8, 6) (dual of [2097152, 2097116, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(829, 2097152, F8, 5) (dual of [2097152, 2097123, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(80, 7, F8, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
(30, 36, large)-Net in Base 8 — Upper bound on s
There is no (30, 36, large)-net in base 8, because
- 4 times m-reduction [i] would yield (30, 32, large)-net in base 8, but