Best Known (138−23, 138, s)-Nets in Base 8
(138−23, 138, 47664)-Net over F8 — Constructive and digital
Digital (115, 138, 47664)-net over F8, using
- 81 times duplication [i] based on digital (114, 137, 47664)-net over F8, using
- net defined by OOA [i] based on linear OOA(8137, 47664, F8, 23, 23) (dual of [(47664, 23), 1096135, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8137, 524305, F8, 23) (dual of [524305, 524168, 24]-code), using
- 1 times code embedding in larger space [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
- 1 times code embedding in larger space [i] based on linear OA(8136, 524304, F8, 23) (dual of [524304, 524168, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8137, 524305, F8, 23) (dual of [524305, 524168, 24]-code), using
- net defined by OOA [i] based on linear OOA(8137, 47664, F8, 23, 23) (dual of [(47664, 23), 1096135, 24]-NRT-code), using
(138−23, 138, 597861)-Net over F8 — Digital
Digital (115, 138, 597861)-net over F8, using
(138−23, 138, large)-Net in Base 8 — Upper bound on s
There is no (115, 138, large)-net in base 8, because
- 21 times m-reduction [i] would yield (115, 117, large)-net in base 8, but