Best Known (36−5, 36, s)-Nets in Base 7
(36−5, 36, 5764812)-Net over F7 — Constructive and digital
Digital (31, 36, 5764812)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 2, 8)-net over F7, using
- digital (29, 34, 5764804)-net over F7, using
- trace code for nets [i] based on digital (12, 17, 2882402)-net over F49, using
- net defined by OOA [i] based on linear OOA(4917, 2882402, F49, 5, 5) (dual of [(2882402, 5), 14411993, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(4917, 5764805, F49, 5) (dual of [5764805, 5764788, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(4917, 5764801, F49, 5) (dual of [5764801, 5764784, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(4913, 5764801, F49, 4) (dual of [5764801, 5764788, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(490, 4, F49, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(4917, 5764805, F49, 5) (dual of [5764805, 5764788, 6]-code), using
- net defined by OOA [i] based on linear OOA(4917, 2882402, F49, 5, 5) (dual of [(2882402, 5), 14411993, 6]-NRT-code), using
- trace code for nets [i] based on digital (12, 17, 2882402)-net over F49, using
(36−5, 36, large)-Net over F7 — Digital
Digital (31, 36, large)-net over F7, using
- 72 times duplication [i] based on digital (29, 34, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(734, large, F7, 5) (dual of [large, large−34, 6]-code), using
- trace code [i] based on linear OA(4917, 5764802, F49, 5) (dual of [5764802, 5764785, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- trace code [i] based on linear OA(4917, 5764802, F49, 5) (dual of [5764802, 5764785, 6]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(734, large, F7, 5) (dual of [large, large−34, 6]-code), using
(36−5, 36, large)-Net in Base 7 — Upper bound on s
There is no (31, 36, large)-net in base 7, because
- 3 times m-reduction [i] would yield (31, 33, large)-net in base 7, but