MinT - Best Known (128−13, 128, s)-Nets in Base 8

Best Known (128−13, 128, s)-Nets in Base 8

(128−13, 128, 2807189)-Net over F₈ — Constructive and digital

Digital (115, 128, 2807189)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (24, 30, 10989)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 65)-net over F₈, using
      - net defined by OOA [i] based on linear OOA(8⁴, 65, F₈, 3, 3) (dual of [(65, 3), 191, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(8⁴, 65, F₈, 2, 3) (dual of [(65, 2), 126, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(8⁴, 65, F₈, 3) (dual of [65, 61, 4]-code or 65-cap in PG(3,8)), using
        ovoid in PG(3,Â 8) [i]
    2. digital (20, 26, 10924)-net over F₈, using
      - net defined by OOA [i] based on linear OOA(8²⁶, 10924, F₈, 6, 6) (dual of [(10924, 6), 65518, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(8²⁶, 32772, F₈, 6) (dual of [32772, 32746, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(8²⁶, 32773, F₈, 6) (dual of [32773, 32747, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(8²⁶, 32768, F₈, 6) (dual of [32768, 32742, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(8²¹, 32768, F₈, 5) (dual of [32768, 32747, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(8⁰, 5, F₈, 0) (dual of [5, 5, 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
        OA with only one run / dual of code without redundancy [i]
2. digital (85, 98, 2796200)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁹⁸, 2796200, F₈, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(8⁹⁸, 8388601, F₈, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁹⁸, 8388602, F₈, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
        trace code [i] based on linear OOA(64⁴⁹, 4194301, F₆₄, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁴⁹, 8388602, F₆₄, 13) (dual of [8388602, 8388553, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴⁹, large, F₆₄, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(128−13, 128, 2839892)-Net in Base 8 — Constructive

(115, 128, 2839892)-net in base 8, using

base change [i] based on digital (83, 96, 2839892)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (16, 22, 43692)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16²², 43692, F₁₆, 6, 6) (dual of [(43692, 6), 262130, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(16²², 131076, F₁₆, 6) (dual of [131076, 131054, 7]-code), using
        trace code [i] based on linear OA(256¹¹, 65538, F₂₅₆, 6) (dual of [65538, 65527, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(256¹¹, 65536, F₂₅₆, 6) (dual of [65536, 65525, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(256⁹, 65536, F₂₅₆, 5) (dual of [65536, 65527, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
  2. digital (61, 74, 2796200)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16⁷⁴, 2796200, F₁₆, 14, 13) (dual of [(2796200, 14), 39146726, 14]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OOA(16⁷⁴, 8388601, F₁₆, 2, 13) (dual of [(8388601, 2), 16777128, 14]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁷⁴, 8388602, F₁₆, 2, 13) (dual of [(8388602, 2), 16777130, 14]-NRT-code), using
        trace code [i] based on linear OOA(256³⁷, 4194301, F₂₅₆, 2, 13) (dual of [(4194301, 2), 8388565, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256³⁷, 8388602, F₂₅₆, 13) (dual of [8388602, 8388565, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁷, large, F₂₅₆, 13) (dual of [large, large−37, 14]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(128−13, 128, large)-Net over F₈ — Digital

Digital (115, 128, large)-net over F₈, using

t-expansion [i] based on digital (113, 128, large)-net over F₈, using
- 4 times m-reduction [i] based on digital (113, 132, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹³², large, F₈, 19) (dual of [large, large−132, 20]-code), using
    - 3 times code embedding in larger space [i] based on linear OA(8¹²⁹, large, F₈, 19) (dual of [large, large−129, 20]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(128−13, 128, large)-Net in Base 8 — Upper bound on s

There is no (115, 128, large)-net in base 8, because

11 times m-reduction [i] would yield (115, 117, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (128−13, 128, s)-Nets in Base 8

(128−13, 128, 2807189)-Net over F8 — Constructive and digital

(128−13, 128, 2839892)-Net in Base 8 — Constructive

(128−13, 128, large)-Net over F8 — Digital

(128−13, 128, large)-Net in Base 8 — Upper bound on s

(128−13, 128, 2807189)-Net over F₈ — Constructive and digital

(128−13, 128, large)-Net over F₈ — Digital