Best Known (128−11, 128, s)-Nets in Base 5
(128−11, 128, 3371067)-Net over F5 — Constructive and digital
Digital (117, 128, 3371067)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (21, 26, 15627)-net over F5, using
- net defined by OOA [i] based on linear OOA(526, 15627, F5, 5, 5) (dual of [(15627, 5), 78109, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(526, 31255, F5, 5) (dual of [31255, 31229, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(526, 31256, F5, 5) (dual of [31256, 31230, 6]-code), using
- trace code [i] based on linear OA(2513, 15628, F25, 5) (dual of [15628, 15615, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(2513, 15625, F25, 5) (dual of [15625, 15612, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(2510, 15625, F25, 4) (dual of [15625, 15615, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(250, 3, F25, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- trace code [i] based on linear OA(2513, 15628, F25, 5) (dual of [15628, 15615, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(526, 31256, F5, 5) (dual of [31256, 31230, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(526, 31255, F5, 5) (dual of [31255, 31229, 6]-code), using
- net defined by OOA [i] based on linear OOA(526, 15627, F5, 5, 5) (dual of [(15627, 5), 78109, 6]-NRT-code), using
- digital (91, 102, 3355440)-net over F5, using
- trace code for nets [i] based on digital (40, 51, 1677720)-net over F25, using
- net defined by OOA [i] based on linear OOA(2551, 1677720, F25, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2551, 8388601, F25, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2551, large, F25, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2551, large, F25, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2551, 8388601, F25, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(2551, 1677720, F25, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- trace code for nets [i] based on digital (40, 51, 1677720)-net over F25, using
- digital (21, 26, 15627)-net over F5, using
(128−11, 128, large)-Net over F5 — Digital
Digital (117, 128, large)-net over F5, using
- 51 times duplication [i] based on digital (116, 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
(128−11, 128, large)-Net in Base 5 — Upper bound on s
There is no (117, 128, large)-net in base 5, because
- 9 times m-reduction [i] would yield (117, 119, large)-net in base 5, but