Best Known (70, 70+16, s)-Nets in Base 8
(70, 70+16, 32769)-Net over F8 — Constructive and digital
Digital (70, 86, 32769)-net over F8, using
- net defined by OOA [i] based on linear OOA(886, 32769, F8, 16, 16) (dual of [(32769, 16), 524218, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(886, 262152, F8, 16) (dual of [262152, 262066, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(886, 262157, F8, 16) (dual of [262157, 262071, 17]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(885, 262144, F8, 17) (dual of [262144, 262059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(873, 262144, F8, 14) (dual of [262144, 262071, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(81, 13, F8, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(886, 262157, F8, 16) (dual of [262157, 262071, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(886, 262152, F8, 16) (dual of [262152, 262066, 17]-code), using
(70, 70+16, 262157)-Net over F8 — Digital
Digital (70, 86, 262157)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(886, 262157, F8, 16) (dual of [262157, 262071, 17]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(885, 262144, F8, 17) (dual of [262144, 262059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(873, 262144, F8, 14) (dual of [262144, 262071, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(81, 13, F8, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
(70, 70+16, large)-Net in Base 8 — Upper bound on s
There is no (70, 86, large)-net in base 8, because
- 14 times m-reduction [i] would yield (70, 72, large)-net in base 8, but