Best Known (32−9, 32, s)-Nets in Base 128
(32−9, 32, 532481)-Net over F128 — Constructive and digital
Digital (23, 32, 532481)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (3, 7, 8193)-net over F128, using
- net defined by OOA [i] based on linear OOA(1287, 8193, F128, 4, 4) (dual of [(8193, 4), 32765, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(1287, 16386, F128, 4) (dual of [16386, 16379, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(1287, 16384, F128, 4) (dual of [16384, 16377, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1285, 16384, F128, 3) (dual of [16384, 16379, 4]-code or 16384-cap in PG(4,128)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(1287, 16386, F128, 4) (dual of [16386, 16379, 5]-code), using
- net defined by OOA [i] based on linear OOA(1287, 8193, F128, 4, 4) (dual of [(8193, 4), 32765, 5]-NRT-code), using
- digital (16, 25, 524288)-net over F128, using
- net defined by OOA [i] based on linear OOA(12825, 524288, F128, 9, 9) (dual of [(524288, 9), 4718567, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using
- net defined by OOA [i] based on linear OOA(12825, 524288, F128, 9, 9) (dual of [(524288, 9), 4718567, 10]-NRT-code), using
- digital (3, 7, 8193)-net over F128, using
(32−9, 32, 2097150)-Net in Base 128 — Constructive
(23, 32, 2097150)-net in base 128, using
- base change [i] based on digital (19, 28, 2097150)-net over F256, using
- 2563 times duplication [i] based on digital (16, 25, 2097150)-net over F256, using
- net defined by OOA [i] based on linear OOA(25625, 2097150, F256, 9, 9) (dual of [(2097150, 9), 18874325, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(25625, 8388601, F256, 9) (dual of [8388601, 8388576, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(25625, 8388601, F256, 9) (dual of [8388601, 8388576, 10]-code), using
- net defined by OOA [i] based on linear OOA(25625, 2097150, F256, 9, 9) (dual of [(2097150, 9), 18874325, 10]-NRT-code), using
- 2563 times duplication [i] based on digital (16, 25, 2097150)-net over F256, using
(32−9, 32, 7956580)-Net over F128 — Digital
Digital (23, 32, 7956580)-net over F128, using
(32−9, 32, large)-Net in Base 128
(23, 32, large)-net in base 128, using
- base change [i] based on digital (19, 28, large)-net over F256, using
- 2561 times duplication [i] based on digital (18, 27, large)-net over F256, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25627, large, F256, 9) (dual of [large, large−27, 10]-code), using
- 2 times code embedding in larger space [i] based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 2 times code embedding in larger space [i] based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25627, large, F256, 9) (dual of [large, large−27, 10]-code), using
- 2561 times duplication [i] based on digital (18, 27, large)-net over F256, using
(32−9, 32, large)-Net in Base 128 — Upper bound on s
There is no (23, 32, large)-net in base 128, because
- 7 times m-reduction [i] would yield (23, 25, large)-net in base 128, but