Best Known (88, 88+16, s)-Nets in Base 9
(88, 88+16, 597875)-Net over F9 — Constructive and digital
Digital (88, 104, 597875)-net over F9, using
- 91 times duplication [i] based on digital (87, 103, 597875)-net over F9, using
- net defined by OOA [i] based on linear OOA(9103, 597875, F9, 16, 16) (dual of [(597875, 16), 9565897, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(9103, 4783000, F9, 16) (dual of [4783000, 4782897, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(9103, 4783001, F9, 16) (dual of [4783001, 4782898, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- linear OA(999, 4782969, F9, 16) (dual of [4782969, 4782870, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(971, 4782969, F9, 12) (dual of [4782969, 4782898, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(15) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(9103, 4783001, F9, 16) (dual of [4783001, 4782898, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(9103, 4783000, F9, 16) (dual of [4783000, 4782897, 17]-code), using
- net defined by OOA [i] based on linear OOA(9103, 597875, F9, 16, 16) (dual of [(597875, 16), 9565897, 17]-NRT-code), using
(88, 88+16, 4783003)-Net over F9 — Digital
Digital (88, 104, 4783003)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9104, 4783003, F9, 16) (dual of [4783003, 4782899, 17]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9103, 4783001, F9, 16) (dual of [4783001, 4782898, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- linear OA(999, 4782969, F9, 16) (dual of [4782969, 4782870, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(971, 4782969, F9, 12) (dual of [4782969, 4782898, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(15) ⊂ Ce(11) [i] based on
- linear OA(9103, 4783002, F9, 15) (dual of [4783002, 4782899, 16]-code), using Gilbert–Varšamov bound and bm = 9103 > Vbs−1(k−1) = 165443 597609 647893 113599 597270 670656 805468 819771 675350 475869 186864 992491 664042 988548 512222 654409 [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(9103, 4783001, F9, 16) (dual of [4783001, 4782898, 17]-code), using
- construction X with Varšamov bound [i] based on
(88, 88+16, large)-Net in Base 9 — Upper bound on s
There is no (88, 104, large)-net in base 9, because
- 14 times m-reduction [i] would yield (88, 90, large)-net in base 9, but