Best Known (138−31, 138, s)-Nets in Base 8
(138−31, 138, 2185)-Net over F8 — Constructive and digital
Digital (107, 138, 2185)-net over F8, using
- 81 times duplication [i] based on digital (106, 137, 2185)-net over F8, using
- net defined by OOA [i] based on linear OOA(8137, 2185, F8, 31, 31) (dual of [(2185, 31), 67598, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(8137, 32776, F8, 31) (dual of [32776, 32639, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(8137, 32779, F8, 31) (dual of [32779, 32642, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(8136, 32768, F8, 31) (dual of [32768, 32632, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(8126, 32768, F8, 29) (dual of [32768, 32642, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(81, 11, F8, 1) (dual of [11, 10, 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(8137, 32779, F8, 31) (dual of [32779, 32642, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(8137, 32776, F8, 31) (dual of [32776, 32639, 32]-code), using
- net defined by OOA [i] based on linear OOA(8137, 2185, F8, 31, 31) (dual of [(2185, 31), 67598, 32]-NRT-code), using
(138−31, 138, 30768)-Net over F8 — Digital
Digital (107, 138, 30768)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8138, 30768, F8, 31) (dual of [30768, 30630, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(8138, 32781, F8, 31) (dual of [32781, 32643, 32]-code), using
- construction XX applied to Ce(30) ⊂ Ce(28) ⊂ Ce(27) [i] based on
- linear OA(8136, 32768, F8, 31) (dual of [32768, 32632, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(8126, 32768, F8, 29) (dual of [32768, 32642, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(8121, 32768, F8, 28) (dual of [32768, 32647, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(81, 12, F8, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(30) ⊂ Ce(28) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(8138, 32781, F8, 31) (dual of [32781, 32643, 32]-code), using
(138−31, 138, large)-Net in Base 8 — Upper bound on s
There is no (107, 138, large)-net in base 8, because
- 29 times m-reduction [i] would yield (107, 109, large)-net in base 8, but