Best Known (131−28, 131, s)-Nets in Base 9
(131−28, 131, 4220)-Net over F9 — Constructive and digital
Digital (103, 131, 4220)-net over F9, using
- 93 times duplication [i] based on digital (100, 128, 4220)-net over F9, using
- net defined by OOA [i] based on linear OOA(9128, 4220, F9, 28, 28) (dual of [(4220, 28), 118032, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(9128, 59080, F9, 28) (dual of [59080, 58952, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(9128, 59081, F9, 28) (dual of [59081, 58953, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- linear OA(9121, 59049, F9, 28) (dual of [59049, 58928, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(996, 59049, F9, 22) (dual of [59049, 58953, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(97, 32, F9, 5) (dual of [32, 25, 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(27) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(9128, 59081, F9, 28) (dual of [59081, 58953, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(9128, 59080, F9, 28) (dual of [59080, 58952, 29]-code), using
- net defined by OOA [i] based on linear OOA(9128, 4220, F9, 28, 28) (dual of [(4220, 28), 118032, 29]-NRT-code), using
(131−28, 131, 59090)-Net over F9 — Digital
Digital (103, 131, 59090)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9131, 59090, F9, 28) (dual of [59090, 58959, 29]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9130, 59088, F9, 28) (dual of [59088, 58958, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(20) [i] based on
- linear OA(9121, 59049, F9, 28) (dual of [59049, 58928, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(991, 59049, F9, 21) (dual of [59049, 58958, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(99, 39, F9, 6) (dual of [39, 30, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(99, 41, F9, 6) (dual of [41, 32, 7]-code), using
- a “GraCyc†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(99, 41, F9, 6) (dual of [41, 32, 7]-code), using
- construction X applied to Ce(27) ⊂ Ce(20) [i] based on
- linear OA(9130, 59089, F9, 27) (dual of [59089, 58959, 28]-code), using Gilbert–Varšamov bound and bm = 9130 > Vbs−1(k−1) = 8 537623 522245 579710 858375 579730 972001 592903 076995 397879 098520 791818 555281 521693 535508 683948 336209 605356 515944 973310 827649 [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(9130, 59088, F9, 28) (dual of [59088, 58958, 29]-code), using
- construction X with Varšamov bound [i] based on
(131−28, 131, large)-Net in Base 9 — Upper bound on s
There is no (103, 131, large)-net in base 9, because
- 26 times m-reduction [i] would yield (103, 105, large)-net in base 9, but