On a duality for codes over non-abelian groups

Abstract

This work is motivated by a well-known open problem in coding theory, asking whether there is a duality theory for codes over non-abelian groups, see Dougherty et al. (Contemp Math 634:79–99, 2015). We prove that such a duality cannot be induced by a duality of a group lattice, and then study a variation that reduces to a group theoretic investigation: We say a finite group of order m has a layer-symmetric lattice if for every divisor d of m there is a bijection between the subgroups of order d and the subgroups of order m/d. We prove that every such group is nilpotent, and then investigate the class of finite p-groups with a layer-symmetric lattice.

Appendices

Appendix A: MacWilliams identity for codes over abelian groups

We recall the theory of the MacWilliams identity for codes over abelian groups. We start by reviewing the theory of characters of finite abelian groups; a good and complete account can be found in [4]. It was hard to find a complete picture including the coding theoretic aspect in the literature, therefore we decided to include this Appendix. As nothing in this section apart from the write-up is our original work, we have omitted the proofs. Throughout this section, let G be a finite abelian group and denote by $\widehat{G}$ the character group of G, that is, $\widehat{G}=\mathrm {Hom}_\mathbb {Z}(G,\mathbb {C}^\star )$ is the group of all homomorphisms $G\rightarrow \mathbb {C}^\star$ with pointwise multiplication. While G and $\widehat{G}$ are isomorphic (with non-canonical isomorphism), the evaluation map $g\mapsto (\chi \mapsto \chi (g))$ describes a canonical isomorphism between G and $\widehat{\widehat{G}}$. For a subgroup $H\le G$ define

$$\begin{aligned} H^\perp =\left\{ \chi \in \widehat{G} : \chi (h)=1\text { for all } h\in H \right\} . \end{aligned}$$

Lemma 22

If $H\le G$ is a subgroup, then the set $H^\perp$ is a subgroup of $\widehat{G}$ isomorphic to $\widehat{G/H}$, the group $\widehat{H}$ is isomorphic to $\widehat{G}/H^\perp$, and $|G|=|H||H^\perp |$ and $(H^\perp )^\perp =H$.

By what is shown above, the map $\mathcal {L}(G)\rightarrow \mathcal {L}(\widehat{G})$, $H\mapsto H^\perp$, describes a bijection between the subgroups of G and the subgroups of $\widehat{G}$ such that if $A,B\le G$, then $A\le B$ if and only if $B^\perp \le A^\perp$; in other words, $\perp$ describes a duality of $\mathcal {L}(G)$. Let V be a complex vector space. For a map $f:G\rightarrow V$ define

$$\begin{aligned} \widehat{f}:\widehat{G}\rightarrow V,\quad \chi \mapsto \sum \nolimits _{x\in G} \chi (x)f(x). \end{aligned}$$

For $\chi \in \widehat{G}$ let $\bar{\chi }$ be the conjugate character, so $\bar{\chi }(x)=\chi (x)^{-1}=\chi (x^{-1})$. Then for all $g\in G$ we have

$$\begin{aligned} f(g)=\tfrac{1}{|G|}\sum \nolimits _{\chi \in \widehat{G}} \bar{\chi }(g)\widehat{f}(\chi ). \end{aligned}$$

Proposition 23

If $H\le G$ and $f:G\rightarrow V$, then for every $g\in G$ we have

$$\begin{aligned} \tfrac{1}{|H|}\sum \nolimits _{h\in H} f(gh)=\tfrac{1}{|G|}\sum \nolimits _{\chi \in H^\perp } \bar{\chi }(g)\widehat{f}(\chi ); \end{aligned}$$

for $g=1$ this yields $\tfrac{1}{|H|}\sum \nolimits _{h\in H} f(h)=\tfrac{1}{|G|}\sum \nolimits _{\chi \in H^\perp } \widehat{f}(\chi )$.

Given a finite abelian group G, define its dual abelian group as $\mathrm {Hom}_\mathbb {Z}(G,\mathbb {Q} / \mathbb {Z})$. This is an additive group with the identity element being the zero homomorphism from G to $\mathbb {Q} / \mathbb {Z}$. The group homomorphism $\mathbb {Q}/\mathbb {Z}\rightarrow \mathbb {C}$, $x\mapsto \exp (2\pi \imath x)$, is injective and its image consists of the elements of finite order in $\mathbb {C}^\star$. This induces a homomorphism $\beta :\mathrm {Hom}_\mathbb {Z}(G,\mathbb {Q}/\mathbb {Z}) \rightarrow \widehat{G} = \mathrm {Hom}_\mathbb {Z}(G,\mathbb {C}^\star ),$ which, if G is finite, induces an isomorphism $\mathrm {Hom}_\mathbb {Z}(G,\mathbb {Q}/\mathbb {Z}) \cong \widehat{G}$. Convention: Below we will use the correspondence between the multiplicative character $\phi \in \widehat{G}$ and the additive character $\phi '\in \mathrm {Hom}_\mathbb {Z}(G,\mathbb {Q}/\mathbb {Z})$ where $\phi =\exp (2\pi \imath {\phi '})$.

Let $C\le G^n$ be a linear code of length n over the finite abelian group G. We write $g\in G^n$ as $g=(g_1,\ldots ,g_n)$ with each $g_i\in G$; the Hamming weight of g is defined as $\mathrm {wt}(g)=|\{i\in \{1,\ldots ,n\} : g_i\ne 0\}|$. Let $V=\mathbb {C}[\mathtt {x},\mathtt {y}]$ be the complex vector space of polynomials in two variables, and define

$$\begin{aligned} W_C(\mathtt {x},\mathtt {y})=\sum \nolimits _{c\in C} \mathtt {x}^{n-\mathrm {wt}(c)}\mathtt {y}^{\mathrm {wt}(c)}=\sum \nolimits _{j=0}^n a_j \mathtt {x}^{n-j}\mathtt {y}^j; \end{aligned}$$

thus, $a_j$ is the number of elements in C of Hamming weight j.

Theorem 24

(MacWilliams Identity) We have $W_{C^\perp }(\mathtt {x},\mathtt {y})=\tfrac{1}{|C|}W_C(\mathtt {x}+(|G|-1)\mathtt {y},\mathtt {x}-\mathtt {y})$.

The following technical proposition is at the heart of the proof:

Proposition 25

Let $\mathcal {V}$ be a commutative complex algebra. If $f:G^n\rightarrow \mathcal {V}$ has the form

$$\begin{aligned} f(x_1,\ldots ,x_n)=\prod \nolimits _{i=1}^n f_i(x_i) \end{aligned}$$

with each $f_i:G\rightarrow \mathcal {V}$, then $\widehat{f}=\prod _{i=1}^n \widehat{f_i}$: if $\phi =(\phi _1,\ldots ,\phi _n)\in \widehat{G^n}\cong \widehat{G}^n$, then $\widehat{f}(\phi )=\prod \nolimits _{i=1}^n \widehat{f_i}(\phi _i)$.

The next lemma studies the individual pieces $\widehat{f_i}(\phi _i)$:

Lemma 26

Let $f_i:G\rightarrow V$, $x\mapsto \mathtt {x}^{1-\mathrm {wt}(x)}\mathtt {y}^{\mathrm {wt}(x)}$, and let $\phi =(\phi _1,\ldots ,\phi _n)\in \widehat{G}^n$. For each i we have

$$\begin{aligned} \widehat{f}_i(\phi _i)= \left\{ \begin{array}{ll} \mathtt {x}+(|G|-1)\mathtt {y}&{} \text{ if } \phi _i=1\; (\text {that is, if }\phi '_i=0)\\ \mathtt {x}-\mathtt {y}&{} \text{ if } \phi _i\ne 1\; (\text {that is, if }\phi _i'\ne 0); \\ \end{array} \right. \end{aligned}$$

thus, $\widehat{f}(\phi )= (\mathtt {x}+(|G|-1)\mathtt {y})^{n-\mathrm {wt}{(\phi ')}} (\mathtt {x}-\mathtt {y})^{\mathrm {wt}{(\phi ')}}$, where $\mathrm {wt}(\phi ')$ is the number of entries $\phi '_i\ne 0$.

Proof of Theorem 24

Use $f:G^n\rightarrow V$, $g\mapsto \mathtt {x}^{n-\mathrm {wt}{(g)}}\mathtt {y}^{\mathrm {wt}{(g)}}$ as before. Using Proposition 23 and Lemma 26, we obtain

$$\begin{aligned} W_C(\mathtt {x},\mathtt {y})= & {} \sum \nolimits _{x\in C} f(x) = \tfrac{|C|}{|G|^n} \sum \nolimits _{\phi \in C^\perp } \widehat{f}(\phi )\\= & {} \tfrac{|C|}{|G|^n} \sum _{\phi \in C^\perp } (\mathtt {x}+(|G|-1)\mathtt {y})^{n-\mathrm {wt}{(\phi ')}} (\mathtt {x}-\mathtt {y})^{\mathrm {wt}{(\phi ')}}\\= & {} \tfrac{|C|}{|G|^n} W_{C^\perp }(\mathtt {x}+(|G|-1)\mathtt {y},\mathtt {x}-\mathtt {y}). \end{aligned}$$

Changing the roles of C and $C^\perp$, we obtain

$$\begin{aligned} W_{C^\perp }(\mathtt {x},\mathtt {y}) = \tfrac{|C^\perp |}{|G|^n} W_{(C^\perp )^\perp }(\mathtt {x}+(|G|-1)\mathtt {y},\mathtt {x}-\mathtt {y})=\tfrac{1}{|C|} W_{C}(\mathtt {x}+(|G|-1)\mathtt {y},\mathtt {x}-\mathtt {y}). \end{aligned}$$

$\square$

Appendix B: More explicit examples

Explicit computations with the computer algebra system GAP [6] show the following. Note that a group written as Q.N or $Q\ltimes N$ is not uniquely determined up to isomorphism. Recall that Proposition 9 considers groups of order dividing $p^4$, so here we consider larger powers of p. Below we also list the GAP IDs of some groups (as stored in the SmallGroups Library of GAP).

• (a) There are 6 non-abelian groups of order $2^5$ that have a layer-symmetric lattice; of these, 3 groups are not self-dual and they can be described as: $C_{2^3}\ltimes C_{2^2}, C_2\times (C_{2^2}\ltimes C_{2^2}), C_2.(C_{2^3}\times C_2).$ The IDs are $[2^5,x]$ with $x\in \{10,12,23\}$.

• (b) There are 23 non-abelian groups of order $2^6$ that have a layer-symmetric lattice; of these, 15 groups are not self-dual, and they can all be described as one of the following:

  $$\begin{aligned} \begin{array}{llll} C_{2^4}\ltimes C_{2^2}, &{} C_2^2.(C_{2^2}\times C_2^2), &{} C_2^2.C_{2^2}^2,&{} C_2^2.(C_{2^3}\times C_2),\\ C_2.(C_{2^2}^2\times C_2), &{} C_2.(C_{2^3}\times C_2^2),&{} C_{2^3}. C_{2^3}, &{} C_{2^3}\ltimes C_{2^3},\\ C_2^2\times (C_{2^2}{\ltimes } C_{2^2}),&{} C_2^2\times (C_{2^3}\ltimes C_{2^2}), &{} C_2\times ( C_2.(C_{2^3}{\times } C_2)),&{} C_2{\times } (C_{2^3}{\ltimes } C_{2^2}),\\ C_{2^2}\times (C_2\ltimes (C_{2^2}\times C_2)). \end{array} \end{aligned}$$

  The IDs are $[2^6,x]$ with $x\in \{15, 16, 43, 44, 45, 65, 86, 96, 100, 103, 104, 105, 194, 198, 201\}$.

• (c) There are 46 non-abelian groups of order $2^7$ that have a layer-symmetric lattice; of these, 29 groups are not self-dual, and they can all be described as one of the following:

  $$\begin{aligned} \begin{array}{llll} C_2.C_{2^2}^3,&{} C_{2^4}.C_{2^3},&{} C_{2^4}\ltimes C_{2^3},&{} C_2^2 . (C_{2^4}\times C_2),\\ C_2^2.(C_{2^3}{\times } C_2^2), &{} C_2^2.(C_{2^3}{\times } C_{2^2}),&{} C_2.(C_{2^3}{\times } C_{2^2}{\times } C_2), &{} C_2{\ltimes } (C_{2^3}{\times } C_{2^2}{\times } C_2),\\ C_{2^5}\ltimes C_{2^2}, &{}C_2^3\times (C_{2^2}\ltimes C_{2^2}), &{} C_2^2\times (C_2.(C_{2^3}\times C_2)), &{} C_2\times (C_{2^4}\ltimes C_{2^2}),\\ C_2\times ( C_2.(C_{2^2}^2\times C_2)), &{} C_2\times (C_{2^3}. C_{2^3}), &{} C_2\times (C_{2^3}\ltimes C_{2^3}). \end{array} \end{aligned}$$

  The IDs are $[2^7,x]$ with $x\in \{99, 102, 153, 154, 294, 295, 298, 299, 569, 570, 574, 881, 882, 884, 954, 1016, 1623, 1634, 1635, 1679, 1698, 1699, 1700, 1701, 1724, 1725, 1726, 1727, 2152\}$.

• (d) There are 19 non-abelian groups of order $3^5$ that have a layer-symmetric lattice; of these, 2 groups are not self-dual, namely: $C_3^2.C_3^3$ and $C_3\times (C_3.(C_{3^2}\times C_3))$. The IDs are $[3^5,x]$ with $x\in \{54,59\}$.

• (e) There are 29 non-abelian groups of order $3^6$ that have a layer-symmetric lattice; of these, 11 groups are not self-dual, and they can all be described as one of the following:

  $$\begin{aligned} \begin{array}{llll} C_3^2.(C_3^2\times C_{3^2}),&{}C_3^2.C_{3^2}^2, &{} C_3.(C_3^2\times C_{3^3}),&{} C_3\ltimes (C_3\times C_{3^2}^2),\\ C_{3^2}\ltimes (C_{3^3}\times C_3), &{} C_3^2\times (C_3.(C_{3^2}\times C_3)),&{} C_3\times (C_3^2.C_3^3) &{} C_{3^2}\times (C_3\ltimes C_{3^2}\times C_3). \end{array} \end{aligned}$$

  The IDs are $[3^6,x]$ with $x\in \{ 241, 267, 269, 421, 424, 439, 447, 460, 461, 482, 487\}$.

• (f) There are 48 non-abelian groups of order $3^7$ that have a layer-symmetric lattice; of these, 12 groups are not self-dual, and they can be described as: $C_3^2.(C_3^2\times C_{3^3})$, $C_3\ltimes (C_{3^3}\times C_{3^2}\times C_3)$, $C_3^3\times (C_3.(C_3\times C_{3^2}))$, $C_3^2\times (C_3^2.C_3^3)$, $C_3^2.(C_{3^3}\times C_{3^2})$. The IDs are $[3^7,x]$ with $x\in \{8076, 8077, 8078, 8079, 8104, 8105, 8106, 8107, 8108, 8109, 9279, 9284\}$.

• (g) For $p\in \{5,7,11\}$, there are 8 non-abelian groups of order $p^5$ with layer-symmetric lattice, all self-dual.

• (h) For $p\in \{5,7\}$, there are 27 non-abelian groups of order $p^6$ that have a layer-symmetric lattice; of these, 9 are not self-dual, and they can all be described as one of

  $$\begin{aligned} \begin{array}{lll} C_p.(C_p^2\times C_{p^3}),&{} C_p\ltimes (C_{p^3}\times C_{p^2}),&{} C_p\ltimes (C_{p^2}^2\times C_p),\\ C_p^2.(C_p^2\times C_{p^2}),&{} C_{p^2}\ltimes (C_{p^3}\times C_p),&{} C_{p^2}\times (C_p\ltimes (C_{p^2}\times C_p)). \end{array} \end{aligned}$$

  The IDs are $[p^6,x]$ with $x\in \{ 15, 22, 79, 90, 128, 182, 246, 255, 270\}$ for $p=5$ and $x\in \{ 15, 22, 79, 92,$$134, 192, 268, 279, 296\}$ for $p=7$.