Best Known (113, 136, s)-Nets in Base 8
(113, 136, 47663)-Net over F8 — Constructive and digital
Digital (113, 136, 47663)-net over F8, using
- 82 times duplication [i] based on digital (111, 134, 47663)-net over F8, using
- net defined by OOA [i] based on linear OOA(8134, 47663, F8, 23, 23) (dual of [(47663, 23), 1096115, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8134, 524294, F8, 23) (dual of [524294, 524160, 24]-code), using
- trace code [i] based on linear OA(6467, 262147, F64, 23) (dual of [262147, 262080, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(6467, 262144, F64, 23) (dual of [262144, 262077, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(6464, 262144, F64, 22) (dual of [262144, 262080, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- trace code [i] based on linear OA(6467, 262147, F64, 23) (dual of [262147, 262080, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8134, 524294, F8, 23) (dual of [524294, 524160, 24]-code), using
- net defined by OOA [i] based on linear OOA(8134, 47663, F8, 23, 23) (dual of [(47663, 23), 1096115, 24]-NRT-code), using
(113, 136, 524304)-Net over F8 — Digital
Digital (113, 136, 524304)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8136, 524304, F8, 23) (dual of [524304, 524168, 24]-code), using
- trace code [i] based on linear OA(6468, 262152, F64, 23) (dual of [262152, 262084, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(6467, 262145, F64, 23) (dual of [262145, 262078, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(6461, 262145, F64, 21) (dual of [262145, 262084, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- trace code [i] based on linear OA(6468, 262152, F64, 23) (dual of [262152, 262084, 24]-code), using
(113, 136, large)-Net in Base 8 — Upper bound on s
There is no (113, 136, large)-net in base 8, because
- 21 times m-reduction [i] would yield (113, 115, large)-net in base 8, but