Best Known (6, 6+4, s)-Nets in Base 7
(6, 6+4, 174)-Net over F7 — Constructive and digital
Digital (6, 10, 174)-net over F7, using
- net defined by OOA [i] based on linear OOA(710, 174, F7, 4, 4) (dual of [(174, 4), 686, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(710, 174, F7, 3, 4) (dual of [(174, 3), 512, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(710, 348, F7, 4) (dual of [348, 338, 5]-code), using
- construction XX applied to C1 = C([341,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([341,2]) [i] based on
- linear OA(77, 342, F7, 3) (dual of [342, 335, 4]-code or 342-cap in PG(6,7)), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,1}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(77, 342, F7, 3) (dual of [342, 335, 4]-code or 342-cap in PG(6,7)), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(710, 342, F7, 4) (dual of [342, 332, 5]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,1,2}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(74, 342, F7, 2) (dual of [342, 338, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(70, 3, F7, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(70, 3, F7, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([341,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([341,2]) [i] based on
- OA 2-folding and stacking [i] based on linear OA(710, 348, F7, 4) (dual of [348, 338, 5]-code), using
- appending kth column [i] based on linear OOA(710, 174, F7, 3, 4) (dual of [(174, 3), 512, 5]-NRT-code), using
(6, 6+4, 348)-Net over F7 — Digital
Digital (6, 10, 348)-net over F7, using
- net defined by OOA [i] based on linear OOA(710, 348, F7, 4, 4) (dual of [(348, 4), 1382, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(710, 348, F7, 3, 4) (dual of [(348, 3), 1034, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(710, 348, F7, 4) (dual of [348, 338, 5]-code), using
- construction XX applied to C1 = C([341,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([341,2]) [i] based on
- linear OA(77, 342, F7, 3) (dual of [342, 335, 4]-code or 342-cap in PG(6,7)), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,1}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(77, 342, F7, 3) (dual of [342, 335, 4]-code or 342-cap in PG(6,7)), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(710, 342, F7, 4) (dual of [342, 332, 5]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,1,2}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(74, 342, F7, 2) (dual of [342, 338, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(70, 3, F7, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(70, 3, F7, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([341,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([341,2]) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(710, 348, F7, 4) (dual of [348, 338, 5]-code), using
- appending kth column [i] based on linear OOA(710, 348, F7, 3, 4) (dual of [(348, 3), 1034, 5]-NRT-code), using
(6, 6+4, 3960)-Net in Base 7 — Upper bound on s
There is no (6, 10, 3961)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 282 530209 > 710 [i]