Best Known (123, 159, s)-Nets in Base 8
(123, 159, 1821)-Net over F8 — Constructive and digital
Digital (123, 159, 1821)-net over F8, using
- 82 times duplication [i] based on digital (121, 157, 1821)-net over F8, using
- net defined by OOA [i] based on linear OOA(8157, 1821, F8, 36, 36) (dual of [(1821, 36), 65399, 37]-NRT-code), using
- OA 18-folding and stacking [i] based on linear OA(8157, 32778, F8, 36) (dual of [32778, 32621, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(8157, 32779, F8, 36) (dual of [32779, 32622, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(33) [i] based on
- linear OA(8156, 32768, F8, 36) (dual of [32768, 32612, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(8146, 32768, F8, 34) (dual of [32768, 32622, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [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(35) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(8157, 32779, F8, 36) (dual of [32779, 32622, 37]-code), using
- OA 18-folding and stacking [i] based on linear OA(8157, 32778, F8, 36) (dual of [32778, 32621, 37]-code), using
- net defined by OOA [i] based on linear OOA(8157, 1821, F8, 36, 36) (dual of [(1821, 36), 65399, 37]-NRT-code), using
(123, 159, 30394)-Net over F8 — Digital
Digital (123, 159, 30394)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8159, 30394, F8, 36) (dual of [30394, 30235, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(8159, 32786, F8, 36) (dual of [32786, 32627, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(32) [i] based on
- linear OA(8156, 32768, F8, 36) (dual of [32768, 32612, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(8141, 32768, F8, 33) (dual of [32768, 32627, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(83, 18, F8, 2) (dual of [18, 15, 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(35) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(8159, 32786, F8, 36) (dual of [32786, 32627, 37]-code), using
(123, 159, large)-Net in Base 8 — Upper bound on s
There is no (123, 159, large)-net in base 8, because
- 34 times m-reduction [i] would yield (123, 125, large)-net in base 8, but