Best Known (137, 137+23, s)-Nets in Base 8
(137, 137+23, 190685)-Net over F8 — Constructive and digital
Digital (137, 160, 190685)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (8, 19, 35)-net over F8, using
- net from sequence [i] based on digital (8, 34)-sequence over F8, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F8 with g(F) = 7, N(F) = 34, and 1 place with degree 2 [i] based on function field F/F8 with g(F) = 7 and N(F) ≥ 34, using a function field by Sémirat [i]
- net from sequence [i] based on digital (8, 34)-sequence over F8, using
- digital (118, 141, 190650)-net over F8, using
- net defined by OOA [i] based on linear OOA(8141, 190650, F8, 23, 23) (dual of [(190650, 23), 4384809, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8141, 2097151, F8, 23) (dual of [2097151, 2097010, 24]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(8141, 2097152, F8, 23) (dual of [2097152, 2097011, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8141, 2097151, F8, 23) (dual of [2097151, 2097010, 24]-code), using
- net defined by OOA [i] based on linear OOA(8141, 190650, F8, 23, 23) (dual of [(190650, 23), 4384809, 24]-NRT-code), using
- digital (8, 19, 35)-net over F8, using
(137, 137+23, 381301)-Net in Base 8 — Constructive
(137, 160, 381301)-net in base 8, using
- net defined by OOA [i] based on OOA(8160, 381301, S8, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(8160, 4194312, S8, 23), using
- discarding factors based on OA(8160, 4194320, S8, 23), using
- trace code [i] based on OA(6480, 2097160, S64, 23), using
- discarding parts of the base [i] based on linear OA(12868, 2097160, F128, 23) (dual of [2097160, 2097092, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(12867, 2097153, F128, 23) (dual of [2097153, 2097086, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(12861, 2097153, F128, 21) (dual of [2097153, 2097092, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 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
- discarding parts of the base [i] based on linear OA(12868, 2097160, F128, 23) (dual of [2097160, 2097092, 24]-code), using
- trace code [i] based on OA(6480, 2097160, S64, 23), using
- discarding factors based on OA(8160, 4194320, S8, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(8160, 4194312, S8, 23), using
(137, 137+23, 4782810)-Net over F8 — Digital
Digital (137, 160, 4782810)-net over F8, using
(137, 137+23, large)-Net in Base 8 — Upper bound on s
There is no (137, 160, large)-net in base 8, because
- 21 times m-reduction [i] would yield (137, 139, large)-net in base 8, but