Best Known (135, 135+31, s)-Nets in Base 8
(135, 135+31, 17477)-Net over F8 — Constructive and digital
Digital (135, 166, 17477)-net over F8, using
- 82 times duplication [i] based on digital (133, 164, 17477)-net over F8, using
- net defined by OOA [i] based on linear OOA(8164, 17477, F8, 31, 31) (dual of [(17477, 31), 541623, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(8164, 262156, F8, 31) (dual of [262156, 261992, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(8164, 262157, F8, 31) (dual of [262157, 261993, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(8163, 262144, F8, 31) (dual of [262144, 261981, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(8151, 262144, F8, 29) (dual of [262144, 261993, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(8164, 262157, F8, 31) (dual of [262157, 261993, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(8164, 262156, F8, 31) (dual of [262156, 261992, 32]-code), using
- net defined by OOA [i] based on linear OOA(8164, 17477, F8, 31, 31) (dual of [(17477, 31), 541623, 32]-NRT-code), using
(135, 135+31, 229225)-Net over F8 — Digital
Digital (135, 166, 229225)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8166, 229225, F8, 31) (dual of [229225, 229059, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(8166, 262165, F8, 31) (dual of [262165, 261999, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(8163, 262144, F8, 31) (dual of [262144, 261981, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(8145, 262144, F8, 28) (dual of [262144, 261999, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(83, 21, F8, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(8166, 262165, F8, 31) (dual of [262165, 261999, 32]-code), using
(135, 135+31, large)-Net in Base 8 — Upper bound on s
There is no (135, 166, large)-net in base 8, because
- 29 times m-reduction [i] would yield (135, 137, large)-net in base 8, but