Best Known (20−6, 20, s)-Nets in Base 7
(20−6, 20, 269)-Net over F7 — Constructive and digital
Digital (14, 20, 269)-net over F7, using
- net defined by OOA [i] based on linear OOA(720, 269, F7, 6, 6) (dual of [(269, 6), 1594, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(720, 807, F7, 6) (dual of [807, 787, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(720, 808, F7, 6) (dual of [808, 788, 7]-code), using
- construction XX applied to C1 = C([130,134]), C2 = C([129,133]), C3 = C1 + C2 = C([130,133]), and C∩ = C1 ∩ C2 = C([129,134]) [i] based on
- linear OA(716, 800, F7, 5) (dual of [800, 784, 6]-code), using the BCH-code C(I) with length 800 | 74−1, defining interval I = {130,131,132,133,134}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(716, 800, F7, 5) (dual of [800, 784, 6]-code), using the BCH-code C(I) with length 800 | 74−1, defining interval I = {129,130,131,132,133}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(720, 800, F7, 6) (dual of [800, 780, 7]-code), using the BCH-code C(I) with length 800 | 74−1, defining interval I = {129,130,…,134}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(712, 800, F7, 4) (dual of [800, 788, 5]-code), using the BCH-code C(I) with length 800 | 74−1, defining interval I = {130,131,132,133}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 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, 4, F7, 0) (dual of [4, 4, 1]-code) (see above)
- construction XX applied to C1 = C([130,134]), C2 = C([129,133]), C3 = C1 + C2 = C([130,133]), and C∩ = C1 ∩ C2 = C([129,134]) [i] based on
- discarding factors / shortening the dual code based on linear OA(720, 808, F7, 6) (dual of [808, 788, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(720, 807, F7, 6) (dual of [807, 787, 7]-code), using
(20−6, 20, 1046)-Net over F7 — Digital
Digital (14, 20, 1046)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(720, 1046, F7, 6) (dual of [1046, 1026, 7]-code), using
- 694 step Varšamov–Edel lengthening with (ri) = (1, 34 times 0, 1, 111 times 0, 1, 213 times 0, 1, 332 times 0) [i] based on linear OA(716, 348, F7, 6) (dual of [348, 332, 7]-code), using
- construction XX applied to C1 = C([341,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([341,4]) [i] based on
- linear OA(713, 342, F7, 5) (dual of [342, 329, 6]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,1,2,3}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(713, 342, F7, 5) (dual of [342, 329, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(716, 342, F7, 6) (dual of [342, 326, 7]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(710, 342, F7, 4) (dual of [342, 332, 5]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [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([341,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([341,4]) [i] based on
- 694 step Varšamov–Edel lengthening with (ri) = (1, 34 times 0, 1, 111 times 0, 1, 213 times 0, 1, 332 times 0) [i] based on linear OA(716, 348, F7, 6) (dual of [348, 332, 7]-code), using
(20−6, 20, 130381)-Net in Base 7 — Upper bound on s
There is no (14, 20, 130382)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 79794 031111 961545 > 720 [i]