Best Known (121−19, 121, s)-Nets in Base 9
(121−19, 121, 531445)-Net over F9 — Constructive and digital
Digital (102, 121, 531445)-net over F9, using
- 91 times duplication [i] based on digital (101, 120, 531445)-net over F9, using
- net defined by OOA [i] based on linear OOA(9120, 531445, F9, 19, 19) (dual of [(531445, 19), 10097335, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(9120, 4783006, F9, 19) (dual of [4783006, 4782886, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(9120, 4783011, F9, 19) (dual of [4783011, 4782891, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(9113, 4782969, F9, 19) (dual of [4782969, 4782856, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(978, 4782969, F9, 13) (dual of [4782969, 4782891, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(97, 42, F9, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(9120, 4783011, F9, 19) (dual of [4783011, 4782891, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(9120, 4783006, F9, 19) (dual of [4783006, 4782886, 20]-code), using
- net defined by OOA [i] based on linear OOA(9120, 531445, F9, 19, 19) (dual of [(531445, 19), 10097335, 20]-NRT-code), using
(121−19, 121, 4783013)-Net over F9 — Digital
Digital (102, 121, 4783013)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9121, 4783013, F9, 19) (dual of [4783013, 4782892, 20]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9120, 4783011, F9, 19) (dual of [4783011, 4782891, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(9113, 4782969, F9, 19) (dual of [4782969, 4782856, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(978, 4782969, F9, 13) (dual of [4782969, 4782891, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(97, 42, F9, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(9120, 4783012, F9, 18) (dual of [4783012, 4782892, 19]-code), using Gilbert–Varšamov bound and bm = 9120 > Vbs−1(k−1) = 2 271812 905176 899675 836476 567898 270882 884557 464482 716637 110107 173267 609009 189110 542615 111122 712045 170525 909752 398809 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(9120, 4783011, F9, 19) (dual of [4783011, 4782891, 20]-code), using
- construction X with Varšamov bound [i] based on
(121−19, 121, large)-Net in Base 9 — Upper bound on s
There is no (102, 121, large)-net in base 9, because
- 17 times m-reduction [i] would yield (102, 104, large)-net in base 9, but