Best Known (125, 125+25, s)-Nets in Base 8
(125, 125+25, 174763)-Net over F8 — Constructive and digital
Digital (125, 150, 174763)-net over F8, using
- 81 times duplication [i] based on digital (124, 149, 174763)-net over F8, using
- net defined by OOA [i] based on linear OOA(8149, 174763, F8, 25, 25) (dual of [(174763, 25), 4368926, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8149, 2097157, F8, 25) (dual of [2097157, 2097008, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8149, 2097160, F8, 25) (dual of [2097160, 2097011, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(8148, 2097152, F8, 25) (dual of [2097152, 2097004, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(8141, 2097152, F8, 23) (dual of [2097152, 2097011, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(81, 8, F8, 1) (dual of [8, 7, 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(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(8149, 2097160, F8, 25) (dual of [2097160, 2097011, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8149, 2097157, F8, 25) (dual of [2097157, 2097008, 26]-code), using
- net defined by OOA [i] based on linear OOA(8149, 174763, F8, 25, 25) (dual of [(174763, 25), 4368926, 26]-NRT-code), using
(125, 125+25, 1048581)-Net over F8 — Digital
Digital (125, 150, 1048581)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8150, 1048581, F8, 2, 25) (dual of [(1048581, 2), 2097012, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8150, 2097162, F8, 25) (dual of [2097162, 2097012, 26]-code), using
- construction XX applied to Ce(24) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- linear OA(8148, 2097152, F8, 25) (dual of [2097152, 2097004, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(8141, 2097152, F8, 23) (dual of [2097152, 2097011, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(8134, 2097152, F8, 22) (dual of [2097152, 2097018, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(81, 9, F8, 1) (dual of [9, 8, 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(24) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(8150, 2097162, F8, 25) (dual of [2097162, 2097012, 26]-code), using
(125, 125+25, large)-Net in Base 8 — Upper bound on s
There is no (125, 150, large)-net in base 8, because
- 23 times m-reduction [i] would yield (125, 127, large)-net in base 8, but