Best Known (32, 44, s)-Nets in Base 8
(32, 44, 685)-Net over F8 — Constructive and digital
Digital (32, 44, 685)-net over F8, using
- net defined by OOA [i] based on linear OOA(844, 685, F8, 12, 12) (dual of [(685, 12), 8176, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(844, 4110, F8, 12) (dual of [4110, 4066, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(844, 4111, F8, 12) (dual of [4111, 4067, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(841, 4096, F8, 12) (dual of [4096, 4055, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(829, 4096, F8, 9) (dual of [4096, 4067, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(83, 15, F8, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(844, 4111, F8, 12) (dual of [4111, 4067, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(844, 4110, F8, 12) (dual of [4110, 4066, 13]-code), using
(32, 44, 4148)-Net over F8 — Digital
Digital (32, 44, 4148)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(844, 4148, F8, 12) (dual of [4148, 4104, 13]-code), using
- 45 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 37 times 0) [i] based on linear OA(841, 4100, F8, 12) (dual of [4100, 4059, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(841, 4096, F8, 12) (dual of [4096, 4055, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(837, 4096, F8, 11) (dual of [4096, 4059, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- 45 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 37 times 0) [i] based on linear OA(841, 4100, F8, 12) (dual of [4100, 4059, 13]-code), using
(32, 44, 1793837)-Net in Base 8 — Upper bound on s
There is no (32, 44, 1793838)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 5444 524275 825522 785562 747474 397053 994324 > 844 [i]