Best Known (129−10, 129, s)-Nets in Base 5
(129−10, 129, 4332007)-Net over F5 — Constructive and digital
Digital (119, 129, 4332007)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (32, 37, 976567)-net over F5, using
- net defined by OOA [i] based on linear OOA(537, 976567, F5, 5, 5) (dual of [(976567, 5), 4882798, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(537, 1953135, F5, 5) (dual of [1953135, 1953098, 6]-code), using
- 1 times truncation [i] based on linear OA(538, 1953136, F5, 6) (dual of [1953136, 1953098, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(537, 1953125, F5, 6) (dual of [1953125, 1953088, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(528, 1953125, F5, 4) (dual of [1953125, 1953097, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(510, 11, F5, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,5)), using
- dual of repetition code with length 11 [i]
- linear OA(51, 11, F5, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- 1 times truncation [i] based on linear OA(538, 1953136, F5, 6) (dual of [1953136, 1953098, 7]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(537, 1953135, F5, 5) (dual of [1953135, 1953098, 6]-code), using
- net defined by OOA [i] based on linear OOA(537, 976567, F5, 5, 5) (dual of [(976567, 5), 4882798, 6]-NRT-code), using
- digital (82, 92, 3355440)-net over F5, using
- trace code for nets [i] based on digital (36, 46, 1677720)-net over F25, using
- net defined by OOA [i] based on linear OOA(2546, 1677720, F25, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(2546, 8388600, F25, 10) (dual of [8388600, 8388554, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(2546, large, F25, 10) (dual of [large, large−46, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(2546, large, F25, 10) (dual of [large, large−46, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(2546, 8388600, F25, 10) (dual of [8388600, 8388554, 11]-code), using
- net defined by OOA [i] based on linear OOA(2546, 1677720, F25, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- trace code for nets [i] based on digital (36, 46, 1677720)-net over F25, using
- digital (32, 37, 976567)-net over F5, using
(129−10, 129, large)-Net over F5 — Digital
Digital (119, 129, large)-net over F5, using
- 52 times duplication [i] based on digital (117, 127, large)-net over F5, using
- t-expansion [i] based on digital (112, 127, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5127, large, F5, 15) (dual of [large, large−127, 16]-code), using
- 7 times code embedding in larger space [i] based on linear OA(5120, large, F5, 15) (dual of [large, large−120, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 7 times code embedding in larger space [i] based on linear OA(5120, large, F5, 15) (dual of [large, large−120, 16]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5127, large, F5, 15) (dual of [large, large−127, 16]-code), using
- t-expansion [i] based on digital (112, 127, large)-net over F5, using
(129−10, 129, large)-Net in Base 5 — Upper bound on s
There is no (119, 129, large)-net in base 5, because
- 8 times m-reduction [i] would yield (119, 121, large)-net in base 5, but