Best Known (97−15, 97, s)-Nets in Base 9
(97−15, 97, 683285)-Net over F9 — Constructive and digital
Digital (82, 97, 683285)-net over F9, using
- 91 times duplication [i] based on digital (81, 96, 683285)-net over F9, using
- net defined by OOA [i] based on linear OOA(996, 683285, F9, 15, 15) (dual of [(683285, 15), 10249179, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(996, 4782996, F9, 15) (dual of [4782996, 4782900, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(996, 4783001, F9, 15) (dual of [4783001, 4782905, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- linear OA(992, 4782969, F9, 15) (dual of [4782969, 4782877, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(964, 4782969, F9, 11) (dual of [4782969, 4782905, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(94, 32, F9, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,9)), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(996, 4783001, F9, 15) (dual of [4783001, 4782905, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(996, 4782996, F9, 15) (dual of [4782996, 4782900, 16]-code), using
- net defined by OOA [i] based on linear OOA(996, 683285, F9, 15, 15) (dual of [(683285, 15), 10249179, 16]-NRT-code), using
(97−15, 97, 4783003)-Net over F9 — Digital
Digital (82, 97, 4783003)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(997, 4783003, F9, 15) (dual of [4783003, 4782906, 16]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(996, 4783001, F9, 15) (dual of [4783001, 4782905, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- linear OA(992, 4782969, F9, 15) (dual of [4782969, 4782877, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(964, 4782969, F9, 11) (dual of [4782969, 4782905, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(94, 32, F9, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,9)), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- linear OA(996, 4783002, F9, 14) (dual of [4783002, 4782906, 15]-code), using Gilbert–Varšamov bound and bm = 996 > Vbs−1(k−1) = 60532 514037 312329 010299 216538 064110 495898 884390 405791 188430 274537 817394 175196 622687 678409 [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(996, 4783001, F9, 15) (dual of [4783001, 4782905, 16]-code), using
- construction X with Varšamov bound [i] based on
(97−15, 97, large)-Net in Base 9 — Upper bound on s
There is no (82, 97, large)-net in base 9, because
- 13 times m-reduction [i] would yield (82, 84, large)-net in base 9, but