Zero-Error Capacity of Duplication Channels

TL;DR

This paper investigates the zero-error capacity of duplication channels, including models inspired by DNA storage, and constructs optimal codes to achieve error-free communication at maximum rates.

Contribution

It determines the zero-error capacity for duplication channels with various parameters and provides constructions for optimal error-correcting codes.

Findings

Zero-error capacity is explicitly calculated for different duplication parameters.

Optimal codes are constructed for error correction in duplication channels.

The models are relevant to DNA-based data storage systems.

Abstract

This paper is concerned with the problem of error-free communication over the i.i.d. duplication channel which acts on a transmitted sequence $ x_1 \cdots x_n $ by inserting a random number of copies of each symbol $ x_i $ next to the original symbol. The random variables representing the numbers of inserted copies at each position $ i $ are independent and take values in $ \{0, 1, \ldots, r\} $, where $ r $ is a fixed parameter. A more general model in which blocks of $ \ell $ consecutive symbols are being duplicated, and which is inspired by DNA-based data storage systems wherein the stored molecules are subject to tandem-duplication mutations, is also analyzed. A construction of optimal codes correcting all patterns of errors of this type is described, and the zero-error capacity of the duplication channel---the largest rate at which information can be transmitted through it in an error-free manner---is determined for each $ \ell $ and $ r $.

Equations

$$\displaystyle\boldsymbol{\tilde{x}}\$$

$$\displaystyle\boldsymbol{\tilde{y}}\$$

$$\displaystyle\boldsymbol{x}\$$

$$\displaystyle\boldsymbol{y}\$$

$$\displaystyle\boldsymbol{\tilde{x}}\$$

$$\displaystyle\boldsymbol{\tilde{y}}\$$

$$\displaystyle\boldsymbol{x}\$$

$$\displaystyle\boldsymbol{y}\$$

$$\mathcal{S}_{q}(n)\coloneqq\big{(}\mathcal{A}_{q}\!\setminus\!\{0\}\big{)}\times\bigcup_{i=0}^{n-1}\mathcal{A}_{q}^{i}.$$

$$\frac{(ri+1)(r\ell +1{)}^j-1}{r}-1=\left(i+\frac{1}{r}\right)(r\ell +1{)}^j-\frac{1}{r}-1$$

$$=i(r\ell +1{)}^j+\ell \left((r\ell +1{)}^{j-1}+(r\ell +1{)}^{j-1}+\cdots +1\right)-1,$$

$${\mathcal{B}}_{q,\ell ,r}$$

$$:=\left\{\sigma 0^{\frac{(ri+1)(r\ell +1{)}^j-1}{r}-1}:1\le \sigma \le q-1,1\le i\le \ell ,j\ge 0\right\}.$$

$${\mathcal{B}}_{q,1,r}=\left\{\sigma 0^{\frac{(r+1{)}^j-1}{r}-1}:1\le \sigma \le q-1,j\ge 1\right\},$$

$${\mathcal{B}}_{2,1,1}=\left\{10^{2^j-2}:j\ge 1\right\}.$$

$${\mathcal{C}}_{q,\ell ,r}(n):={\mathcal{B}}_{q,\ell ,r}^{\ast }\cap {\mathcal{S}}_q(n),$$

$${\mathcal{C}}_{q,r,\ell }^{\prime }(n)=\left({\mathcal{A}}_q^{\ell }\times {\tilde{\mathcal{B}}}_{q,r,\ell }\times {\mathcal{B}}_{q,r,\ell }^{\ast }\right)\cap \bigcup _{i=0}^n{\mathcal{A}}_q^i,$$

$${\tilde{\mathcal{B}}}_{q,\ell ,r}:=\left\{0^{\frac{(ri+1)(r\ell +1{)}^j-1}{r}-1}:1\le i\le \ell ,j\ge 0\right\}.$$

$$f(\sigma 0^u)=\sigma 0^{\frac{(ri+1)(r\ell +1{)}^j-1}{r}-1},$$

$$i\equiv u+1(\mathrm{mod}\ell ),$$

$$j=\lfloor {\log }_{r\ell +1}\left(\frac{r(u+1)+1}{ri+1}\right)\rfloor .$$

$$v_{q,\ell ,r}(x):=(q-1)\sum _{j=0}^{\infty }\sum _{i=1}^{\ell }{x}^{\frac{(ri+1)(r\ell +1{)}^j-1}{r}}.$$

$$v_{q,1,r}(x)=(q-1)\sum _{j=1}^{\infty }{x}^{\frac{(r+1{)}^j-1}{r}},$$

$$v_{2,1,1}(x)=\sum _{j=1}^{\infty }{x}^{2^j-1}.$$

$$\displaystyle\big{|}\mathcal{C}_{q,\ell,r}(n)\big{|}$$

$$=(q-1)\sum _{j=0}^{\infty }\sum _{i=1}^{\ell }\left|{\mathcal{C}}_{q,\ell ,r}\left(n-\frac{(ri+1)(r\ell +1{)}^j-1}{r}\right)\right|,$$

$$1-v_{q,\ell ,r}(x)=0.$$

$$\mathcal{B}_{q,\ell,\infty}\coloneqq\Big{\{}\sigma\,0^{i}:1\leq\sigma\leq q-1,0\leq i\leq\ell-1\Big{\}},$$

$$v_{q,\ell ,\infty }(x):=(q-1)\sum _{i=1}^{\ell }{x}^i.$$

$$C_0^{\text{0-ins}}(\omega ;q,\ell ,r)=\omega \log \left((q-1)\sum _{j=0}^{\infty }\sum _{i=1}^{\ell }{\rho }_{\omega }^{\frac{(ri+1)(r\ell +1{)}^j-1}{r}-\frac{1}{\omega }}\right),$$

$${\omega }^{\ast }$$

$$={\left((q-1)\sum _{j=0}^{\infty }\sum _{i=1}^{\ell }\frac{(ri+1)(r\ell +1{)}^j-1}{r}{\rho }^{\frac{(ri+1)(r\ell +1{)}^j-1}{r}}\right)}^{-1}$$

Full text

I Introduction and Preliminaries

Channels with synchronization errors are a class of communication models that attempt to capture the impairments in the received signal caused by the loss of synchronization at the symbol level and similar phenomena. Starting with Levenshtein’s work on the insertion/deletion channel [16], this and related models have inspired a significant amount of research due to both their practical relevance and theoretical appeal (see, e.g., the surveys [17, 20]). However, their analysis is in general complicated and they have become somewhat notorious for resisting the attempts of researchers to quantify even their basic information-theoretic limits.

The present paper is focused on a particular type of channels with synchronization errors that are usually referred to as duplications, repetitions, or sticky insertions. These channels model the communication scenario in which the receiver is sampling the received (baseband) signal at a rate faster than the transmitter’s clock, causing some of the symbols to be read more than once and thus resulting in a longer string in which some of the transmitted symbols are erroneously repeated several times. For the purpose of generality, we shall in fact consider a model in which blocks of $\ell$ consecutive symbols are being duplicated, for any fixed parameter $\ell\geq 1$. These kinds of impairments have also attracted some interest lately because they correspond to tandem-duplication mutations of DNA sequences [21] and are thus of potential relevance for DNA-based data storage applications [6, 9, 10, 13].

With the above-mentioned information transmission and storage scenarios as motivating examples, we study the problem of reliable communication in the presence of duplication errors. Our results include a construction of optimal codes correcting all possible patterns of duplication errors, and a consequent characterization of the zero-error capacity of duplication channels, which is the largest rate at which information can be transmitted through them in an error-free manner. The methods used in the analysis are direct extensions of the well-known methods for discrete memoryless channels (DMCs) [24]. Nonetheless, the obtained results are interesting in that they provide one of the first examples of a realistic and non-trivial discrete-time channel with synchronization errors for which such a coding theorem has been established. This is also one of the rare cases where zero-error capacity is easier to determine than the Shannon (vanishing-error) capacity.

In the remainder of this section we describe the channel model, introduce basic terminology, and list the relevant literature. Our main results concerning zero-error codes and capacity of duplication channels are presented in Section II, while Section III presents a refinement of these results pertaining to constant-weight codes.

I-A The Channel Model

The channel alphabet is denoted by $\mathcal{A}_{q}\coloneqq\{0,1,\ldots,q-1\}$, and the set of all words over $\mathcal{A}_{q}$ by $\mathcal{A}_{q}^{*}\coloneqq\bigcup_{i=0}^{\infty}\mathcal{A}_{q}^{i}$.

We start with the description of the model in the special case $\ell=1$. Consider the channel that acts on an input string $\boldsymbol{\tilde{x}}=\tilde{x}_{1}\cdots\tilde{x}_{n}$ by inserting a random number of copies of each symbol $\tilde{x}_{i}$ next to the original symbol. More precisely, for every $i=1,\ldots,n$, the channel inserts $c_{i}$ copies of the symbol $\tilde{x}_{i}$ next to it, where it is assumed that $\textnormal{Pr}\{c_{i}=k\}>0$ for $k=0,1,\ldots,r$ and $\sum_{k=0}^{r}\textnormal{Pr}\{c_{i}=k\}=1$, and that the random variables $c_{i}$ and $c_{j}$ are independent for any $i\neq j$. Thus, the maximum number of inserted copies at each position is $r\in\{1,2,\ldots\}$, a fixed parameter111The assumption that the number of inserted copies of each symbol is bounded by a parameter $r<\infty$ is realistic in sticky-insertion channels where such errors occur due to the mismatch between the transmitter’s and the receiver’s clock. Namely, it is reasonable to expect that this mismatch is small and that only a bounded number of copies of a symbol may be inserted at each position. In fact, the case $r=1$, where each input symbol is either duplicated (once) or is left intact, is likely to apply in practice. . The specific values of the probabilities $\textnormal{Pr}\{c_{i}=k\}$ are irrelevant for the problems addressed in this paper, only the assumption that they are all strictly positive for $k=0,1,\ldots,r$ is needed. We refer to the channel just described as the $q$-ary $(1,r)$-duplication channel, where the word $q$-ary is often omitted as the alphabet is understood from the context.

As an example, consider the following input string $\boldsymbol{\tilde{x}}\in\mathcal{A}_{3}^{7}$ and the corresponding string $\boldsymbol{\tilde{y}}\in\mathcal{A}_{3}^{11}$ obtained at the output of the ternary $(1,2)$-duplication channel:

[TABLE]

The inserted duplicates are underlined; the total number of duplication errors that occurred in the channel is $1+2+1=4$.

By using the transformation $\phi:\mathcal{A}_{q}^{*}\to\mathcal{A}_{q}^{*}$, $\boldsymbol{\tilde{x}}\mapsto\boldsymbol{x}$, defined by $x_{i}=\tilde{x}_{i}-\tilde{x}_{i-1}$, $1\leq i\leq n$, where subtraction is performed modulo $q$ and it is understood that $\tilde{x}_{0}=0$, it is easy to see that duplication errors are essentially equivalent to insertions of zeros [3]. For example, for the strings in (1) we have:

[TABLE]

We can thus define the $q$-ary $(1,r)$-[math]-insertion channel as the channel which acts on an input string $\boldsymbol{x}=x_{1}\cdots x_{n}$ by inserting $c_{i}$ zeros after the symbol $x_{i}$ (independently of everything else), where $\textnormal{Pr}\{c_{i}=k\}>0$ for $k=0,1,\ldots,r$ and $\sum_{k=0}^{r}\textnormal{Pr}\{c_{i}=k\}=1$. By using the bijective mapping $\phi$ one can switch between the $(1,r)$-duplication and the $(1,r)$-[math]-insertion channels, as well as between the corresponding codes, so we shall use both of these equivalent descriptions interchangeably.

Consider a string $\boldsymbol{x}=\sigma_{1}\,0^{u_{1}}\,\cdots\,\sigma_{w}\,0^{u_{w}}$, where $\sigma_{i}\in\mathcal{A}_{q}\!\setminus\!\{0\}$ and $0^{u}$ denotes a block of $u$ zeros. This representation is particularly useful for the [math]-insertion channel as this channel affects only the runs of zeros in an input string $\boldsymbol{x}$. A simple but important observation here is that the effect of the channel on the segment $\sigma_{i}\,0^{u_{i}}$ is independent of its effect on any other segment $\sigma_{j}\,0^{u_{j}}$, $i\neq j$. Therefore, the $(1,r)$-[math]-insertion channel (with an additional assumption that the first symbol of every codeword is non-zero) is equivalent to a DMC with alphabet $\{\sigma\,0^{u}:\sigma\in\mathcal{A}_{q}\!\setminus\!\{0\},u\geq 0\}$. Apart from the alphabet of this DMC being infinite, each element of the alphabet has a “cost” assigned to it [19]. Namely, the element $\sigma\,0^{u}$ is assigned a cost of $u+1$, equal to its length when considered as a string over $\mathcal{A}_{q}$, i.e., the cost is the number of symbols from $\mathcal{A}_{q}$ that need to be transmitted in order for this element to be “transmitted” over the equivalent DMC.

Let us now introduce a more general model in which blocks of length $\ell$ are being duplicated, where $\ell\in\{1,2,\ldots\}$ is a fixed parameter. This generalization is primarily of theoretical value but, as mentioned in the introductory part of the paper, it is of potential interest in DNA-based data storage systems wherein the stored molecules are subject to tandem-duplication mutations [10]. Define the $(\ell,r)$-duplication channel as the channel that acts on an input string $\boldsymbol{\tilde{x}}$ by inserting after the symbol $\tilde{x}_{i}$ at most $r$ copies of the substring $\tilde{x}_{i-\ell+1}\cdots\tilde{x}_{i}$, for every $i=\ell,\ldots,n$. For example, consider the following input string $\boldsymbol{\tilde{x}}\in\mathcal{A}_{3}^{7}$ and the corresponding string $\boldsymbol{\tilde{y}}\in\mathcal{A}_{3}^{16}$ obtained at the output of the ternary $(3,2)$-duplication channel:

[TABLE]

The inserted duplicates of blocks of length $\ell=3$ are underlined; the total number of (tandem) duplication errors that occurred in the channel is $1+2=3$. Again, by using a suitable transformation $x_{i}=\phi_{\ell}(\tilde{x}_{i})=\tilde{x}_{i}-\tilde{x}_{i-\ell}$, one can show that this channel is equivalent to the channel which inserts up to $r$ blocks $0^{\ell}$ after the symbol $x_{i}$, for every $i=\ell,\ldots,n$ [10]. For example, for the strings in (1) we have:

[TABLE]

For simplicity, we shall consider a slight variation of the latter model: let $(\ell,r)$-[math]-insertion channel be the channel which inserts up to $r$ blocks $0^{\ell}$ after the symbol $x_{i}$, for every $i=1,\ldots,n$. The difference is only in the possibility to insert blocks $0^{\ell}$ after the first $\ell-1$ symbols, which is irrelevant for the asymptotic analysis (see Remark 2 ahead on how this assumption can be dropped).

I-B Terminology and Conventions

Two strings $\boldsymbol{x},\boldsymbol{y}\in\mathcal{A}_{q}^{*}$ are said to be confusable in a given communication channel if they can produce the same string at the output of that channel. They are said to be non-confusable otherwise. A set of strings $\mathcal{C}\subseteq\mathcal{A}_{q}^{*}$ is said to be a zero-error code [24] for a given channel if every two different codewords $\boldsymbol{x},\boldsymbol{y}\in\mathcal{C}$ are non-confusable. In other words, the requirement is that the code $\mathcal{C}$ is capable of correcting all possible patterns of errors in a given channel.

For technical reasons (see Remark 1 ahead), we shall consider codes whose codewords are of length $\leq\!n$, rather than being exactly equal to $n$, and whose first symbol is non-zero. More precisely, codes will be defined in the space

[TABLE]

For any such code $\mathcal{C}$, its rate is defined in the usual way as $\frac{1}{n}\log|\mathcal{C}|$, where $\log$ is the logarithm to the base $2$. The zero-error capacity of a channel is the $\limsup_{n\to\infty}$ of the rates of optimal zero-error codes in $\mathcal{S}_{q}(n)$. The zero-error capacity of the $q$-ary $(\ell,r)$-duplication channel is denoted by $C_{0}^{\textnormal{dupl}}(q,\ell,r)$. Note that the convention of defining codes in $\mathcal{S}_{q}(n)$, rather than in, e.g., $\mathcal{A}_{q}^{n}$, does not affect the asymptotically achievable rates, i.e., the capacity. It is adopted solely for convenience as it makes the discussion and the proofs “cleaner”.

Another point that should be mentioned in this context is the following. While zero-error codes are defined here by the (usual) requirement that every two different codewords are non-confusable, in the setting where multiple codewords are to be transmitted in succession over a channel another issue arises that must be resolved in order for a code to be called “zero-error”. Namely, due to the facts that the codewords have possibly different lengths, and that the duplication channel changes the lengths of the transmitted sequences, it is not a priory clear how the receiver will be able to delimit the output sequences that correspond to different codewords, and to decode them correctly. In Section II this problem is ignored and it is assumed that only one codeword is transmitted per session. This assumption is justified in some (DNA-based) data storage scenarios where the stored codewords are physically separated. The issue with the consecutive transmission of multiple codewords is discussed briefly in Section III, and a simple solution is presented.

I-C Related Work and Main Results

The binary [math]-insertion channel was first studied in [15], where a construction of codes correcting $t$ insertions of zeros was presented and asymptotic bounds on the cardinality of optimal codes derived, for any fixed $t$ and $n\to\infty$. No restrictions on the locations of inserted zeros were imposed in [15], so one might refer to this channel as the $(1,\infty)$-[math]-insertion channel (i.e., $r=\infty$). Another construction for the same model was later given in [3]. Generalizations of the constructions from [15] and [3] to the case $\ell>1$ were given in [13] and [14], respectively. The best known asymptotic bounds on codes correcting $t$ insertions of blocks $0^{\ell}$, for fixed $t$ and $n\to\infty$, were reported in [13], where the upper bound from [15] (for $\ell=1$) was improved for every $t>2$, and the bounds were also generalized to the case $\ell>1$.

A construction of codes correcting all possible error patterns in the $(\ell,\infty)$-duplication channel, and the consequent characterization of its zero-error capacity, were given in [10]. We note, however, that for $\ell=1$, which is the main motivating model in the present paper, the zero-error capacity of the $(1,\infty)$-duplication channel is trivially $\log(q-1)$, so the mentioned results from [10] are only interesting for $\ell>1$. The zero-error capacity of a model that can be seen as the continuous-time version of the $(1,r)$-duplication channel was determined in [25]. Namely, in [25] a continuous-time channel was studied in which the input waveforms are $\mathcal{A}_{q}$-valued step functions, and in which the duration of each “symbol” $a\in\mathcal{A}_{q}$ (the length of the interval in which the input waveform has a fixed value $a\in\mathcal{A}_{q}$) is changed by a random multiplicative factor. This is analogous to our discrete-time $(1,r)$-duplication channel in which the length of an input run of identical symbols may change from $u$ to $v$, where $u\leq v\leq u(r+1)$.

Our main results are a construction of optimal zero-error codes and a characterization of the zero-error capacity of the $(\ell,r)$-duplication channel, for any fixed $\ell$ and $r$. The method of construction is very similar to those given in [10] and [25], and all of them can in fact be seen as straightforward extensions of the methods Shannon developed for DMCs [24]. Thus the above-mentioned results from [10] can be recovered as a special case of our results by letting $r\to\infty$, while in another special case ($\ell=1$) our results represent discrete-time analogs of those from [25]. In addition to being optimal in terms of their cardinality, the constructed codes have a very simple combinatorial structure based on which an efficient decoding algorithm is obtained. Moreover, and further generalizing our findings, we obtain a characterization of the constant-weight zero-error capacity of the $(\ell,r)$-[math]-insertion channel.

Finally, we mention in this context several more works on duplication channels that are information theoretic in nature, but are concerned with different problems. The Shannon capacity of duplication channels—the largest rate achievable in the vanishing-error, rather than the zero-error regime—was studied extensively [2, 4, 8, 11, 18, 19, 23]. Tight bounds on the capacity were obtained in these works, but the exact value remains elusive even for the most basic models. The expressive power of duplications as a generative process was studied in the recent works [5, 6, 9].

II Error-Free Communication Over

Duplication Channels

This section presents a solution to the problem of error-free communication over the $(\ell,r)$-[math]-insertion channel. More precisely, a construction of optimal zero-error codes, a description of the corresponding decoding algorithm, and a characterization of the zero-error capacity are given. Properties and numerical values of the capacity as a function of channel parameters are also examined.

II-A Optimal Zero-Error Codes

Let $\mathcal{B}_{q,\ell,r}$ be the set of all finite $q$-ary strings consisting of a non-zero symbol followed by a block of zeros of length

[TABLE]

for some $i\in\{1,2,\ldots,\ell\}$ and $j\geq 0$, that is

[TABLE]

In the important special case $\ell=1$ this reduces to

[TABLE]

and further specializing to $q=2$, $r=1$ we get

[TABLE]

Note that the lengths of the blocks from $\mathcal{B}_{q,\ell,r}$ form a geometric progression (shifted for $1/r$) for every fixed $i$, and that these lengths are congruent to $i$ modulo $\ell$ (see (6)).

Define the code $\mathcal{C}_{q,\ell,r}(n)$ as the set of all strings of length $\leq\!n$ that are composed of blocks from $\mathcal{B}_{q,\ell,r}$, namely

[TABLE]

where $\mathcal{S}_{q}(n)$ was defined in (5). Theorem 1 ahead claims that this code is an optimal zero-error code for the $q$-ary $(\ell,r)$-[math]-insertion channel. Before we state and prove it, we describe first the intuition behind the construction and explain how the set of constituent blocks from (LABEL:eq:B) arises as a solution (see also the discussion on constant-weight codes in Section III, Figure 3 in particular).

Recall that, for a transmitted sequence $\boldsymbol{x}=\sigma_{1}\,0^{u_{1}}\,\cdots\,\sigma_{w}\,0^{u_{w}}$, $\sigma_{i}\neq 0$, the [math]-insertion channel acts on the component blocks $\sigma_{i}0^{u_{i}}$ independently of each other (see Section I-A). One may therefore attempt to find an optimal zero-error code $\mathcal{B}_{q,\ell,r}$ in the set of all possible blocks $\{\sigma\,0^{u}:\sigma\in\mathcal{A}_{q}\!\setminus\!\{0\},u\geq 0\}$, and then construct a code $\mathcal{C}_{q,\ell,r}$ for the $(\ell,r)$-[math]-insertion channel by concatenating the blocks from $\mathcal{B}_{q,\ell,r}$ (i.e., define the code in the set $\mathcal{B}_{q,\ell,r}^{*}$). As it turns out, this approach is optimal.

To find an optimal zero-error code $\mathcal{B}_{q,\ell,r}$, we employ a simple greedy algorithm. First note that the channel preserves both the initial non-zero symbol and the length of the input sequence modulo $\ell$ (as it only adds blocks $0^{\ell}$), so we can independently construct codes in each subset of blocks $\{\sigma\,0^{u}:u+1\equiv i\pmod{\ell}\}$, defined for any fixed $\sigma\in\{1,\ldots,q-1\}$ and $i\in\{1,\ldots,\ell\}$. Now, for each of these $(q-1)\ell$ subsets we do the following. List the elements of a subset according to their length. Select the first element on the list as a codeword, and delete from the list all sequences that this codeword can produce at the output of the $(\ell,r)$-[math]-insertion channel. Then select the first element on the remaining list as a codeword, delete all sequences that it can produce at the output of the channel, and so on. Figure 1 illustrates this procedure for $q=2$, $\ell=1$, $r=1$ (there is only $(q-1)\ell=1$ list in this case). The set of sequences that $1\,0^{u}$ can produce at the output of the $(1,1)$-[math]-insertion channel is $\{1\,0^{v}:u\leq v\leq 2u+1\}$, and so the resulting set of codewords is \mathcal{B}_{2,1,1}=\big{\{}1,1\,0^{2},1\,0^{6},1\,0^{14},\ldots\big{\}}=\big{\{}1\,0^{2^{j}-2}:j\geq 1\big{\}}. For general parameters $q,\ell,r$, the set of sequences that $\sigma\,0^{u}$ can produce at the channel output is \big{\{}\sigma\,0^{u},\sigma\,0^{u+\ell},\ldots,\sigma\,0^{u+(u+1)r\ell}\big{\}} (up to $r$ blocks $0^{\ell}$ are inserted after each of the $(u+1)$ symbols of the input sequence). By using this fact it is not difficult to see that, in the subset of blocks $\{\sigma\,0^{u}:u+1\equiv i\pmod{\ell}\}$, the set of codewords obtained by the procedure just described is \big{\{}\sigma\,0^{i-1},\sigma\,0^{i(r\ell+1)+\ell-1},\sigma\,0^{i(r\ell+1)^{2}+\ell(r\ell+1)+\ell-1},\ldots\big{\}}=\big{\{}\sigma\,0^{i(r\ell+1)^{j}+\ell\left((r\ell+1)^{j-1}+(r\ell+1)^{j-1}+\cdots+1\right)-1}:j\geq 0\big{\}}. The union of these sets over all $i$ and $\sigma$ is precisely the code $\mathcal{B}_{q,\ell,r}$ defined in (LABEL:eq:B).

Theorem 1**.**

$\mathcal{C}_{q,r,\ell}(n)$* is a zero-error code for the $(\ell,r)$-[math]-insertion channel. Moreover, every zero-error code $\mathcal{C}\subseteq\mathcal{S}_{q}(n)$ for the $(\ell,r)$-[math]-insertion channel satisfies $|\mathcal{C}|\leq|\mathcal{C}_{q,r,\ell}(n)|$.*

Proof:

To prove the first part of the statement, we need to demonstrate that any two distinct codewords from $\mathcal{C}_{q,\ell,r}(n)$ are non-confusable in the $(\ell,r)$-[math]-insertion channel. Since all codewords are composed of blocks from $\mathcal{B}_{q,\ell,r}$, and since the channel acts independently on the component blocks of a codeword, it is enough to show that any two distinct blocks from $\mathcal{B}_{q,\ell,r}$ are non-confusable. So consider two such blocks, $\boldsymbol{x}=\sigma\,0^{\frac{(ri+1)(r\ell+1)^{j}-1}{r}-1}$ and $\boldsymbol{y}=\sigma\,0^{\frac{(rk+1)(r\ell+1)^{m}-1}{r}-1}$. There are two cases to examine:

1.) $i\neq k$, and

2.) $i=k$but $j<m$ (the case $j>m$ will follow by symmetry).

In the first case, since $|\boldsymbol{x}|\equiv i\pmod{\ell}$ and $|\boldsymbol{y}|\equiv k\pmod{\ell}$, we have $|\boldsymbol{x}|\not\equiv|\boldsymbol{y}|\pmod{\ell}$. Since the channel preserves the lengths of strings modulo $\ell$, we conclude that $\boldsymbol{x}$ and $\boldsymbol{y}$ cannot produce strings of the same length at the channel output and are therefore non-confusable. In the second case, since the maximum number of inserted zeros in $\boldsymbol{x}$ is $|\boldsymbol{x}|r\ell$, the maximum length of the resulting output string is $|\boldsymbol{x}|+|\boldsymbol{x}|r\ell=\linebreak\frac{(ri+1)(r\ell+1)^{j}-1}{r}(r\ell+1)=\frac{(ri+1)(r\ell+1)^{j+1}-1}{r}-\ell<\frac{(ri+1)(r\ell+1)^{m}-1}{r}=|\boldsymbol{y}|.$ As the channel can only increase the length of the transmitted string, we conclude that $\boldsymbol{y}$ can never produce the same strings at the channel output that $\boldsymbol{x}$ can, and therefore, $\boldsymbol{x}$ and $\boldsymbol{y}$ are non-confusable. This proves that $\mathcal{B}_{q,\ell,r}$ is a zero-error code for the $(\ell,r)$-0-insertion channel, which further implies that $\mathcal{C}_{q,\ell,r}(n)$ is a zero-error code as well.

We now prove the second part of the statement which claims that the code $\mathcal{C}_{q,\ell,r}(n)$ is optimal. Let $f:\mathcal{S}_{q}(n)\to\mathcal{S}_{q}(n)$ be a mapping defined as follows: for any string $\boldsymbol{x}\in\mathcal{S}_{q}(n)$, $f(\boldsymbol{x})$ is obtained by removing from each run of zeros in $\boldsymbol{x}$ as many blocks $0^{\ell}$ as is necessary and sufficient in order to obtain a run of zeros whose length is of the form $\frac{(ri+1)(r\ell+1)^{j}-1}{r}-1$, with $1\leq i\leq\ell$ and $j\geq 0$. We claim that $f$ has the following property: if two strings $\boldsymbol{x},\boldsymbol{y}$ are non-confusable in the $(\ell,r)$-0-insertion channel, then their images $f(\boldsymbol{x}),f(\boldsymbol{y})$ are non-confusable as well. Again, in order to show this, we may assume without loss of generality that $\boldsymbol{x}$ and $\boldsymbol{y}$ are of the form $\boldsymbol{x}=\sigma\,0^{u}$, $\boldsymbol{y}=\sigma\,0^{v}$, for some $\sigma\in\mathcal{A}_{q}\setminus\{0\}$, in which case we have $f(\boldsymbol{x})=\sigma\,0^{\frac{(ri+1)(r\ell+1)^{j}-1}{r}-1}$, $f(\boldsymbol{y})=\sigma\,0^{\frac{(rk+1)(r\ell+1)^{m}-1}{r}-1}$. Let also $|\boldsymbol{x}|<|\boldsymbol{y}|$. Now, suppose that $f(\boldsymbol{x}),f(\boldsymbol{y})$ are confusable. Then, since $f(\boldsymbol{x}),f(\boldsymbol{y})\in\mathcal{B}_{q,\ell,r}$, it follows from the first part of the proof that we must have $f(\boldsymbol{x})=f(\boldsymbol{y})$, i.e, $(i,j)=(k,m)$. By the definition of $f$, we then conclude that $|\boldsymbol{x}|\equiv|\boldsymbol{y}|\equiv i\pmod{\ell}$ (because $f$ removes blocks $0^{\ell}$ from its argument and hence preserves its length modulo $\ell$), and that $\frac{(ri+1)(r\ell+1)^{j}-1}{r}\leq|\boldsymbol{x}|<|\boldsymbol{y}|\leq\frac{(ri+1)(r\ell+1)^{j+1}-1}{r}-\ell$ (where the last inequality holds because assuming otherwise, namely $|\boldsymbol{y}|>\frac{(ri+1)(r\ell+1)^{j+1}-1}{r}-\ell$, and using the fact that $|\boldsymbol{y}|\equiv i\pmod{\ell}$, would imply that $|\boldsymbol{y}|\geq\frac{(ri+1)(r\ell+1)^{j+1}-1}{r}$, and hence $m\geq j+1$, contradicting the above-established fact that $m=j$). This further implies that $|\boldsymbol{y}|-|\boldsymbol{x}|$ is a multiple of $\ell$, and that $|\boldsymbol{y}|-|\boldsymbol{x}|\leq\linebreak(ri+1)(r\ell+1)^{j}\ell-\ell=\frac{(ri+1)(r\ell+1)^{j}-1}{r}r\ell\leq|\boldsymbol{x}|r\ell$. This means that the string $\boldsymbol{y}$ can be obtained from the string $\boldsymbol{x}$ by inserting $|\boldsymbol{x}|r$ blocks $0^{\ell}$ in the latter, and therefore $\boldsymbol{x}$ and $\boldsymbol{y}$ are confusable in the $(\ell,r)$-0-insertion channel. We have thus established the claimed property of $f$: if $f(\boldsymbol{x})$ and $f(\boldsymbol{y})$ are confusable, then so are $\boldsymbol{x}$ and $\boldsymbol{y}$. Now let $\mathcal{C}\subseteq\mathcal{S}_{q}(n)$ be an arbitrary zero-error code for the $(\ell,r)$-0-insertion channel, and denote by $f(\mathcal{C})$ the image of $\mathcal{C}$ under the mapping $f$. We first notice that $f(\mathcal{C})$ is itself a zero-error code, because $f(\mathcal{C})\subseteq\mathcal{C}_{q,\ell,r}(n)$ by the definition of $f$. Moreover, $|f(\mathcal{C})|=|\mathcal{C}|$, i.e., $f$ is injective over any zero-error code $\mathcal{C}$. This follows from the above-mentioned property of $f$: since any two different codewords $\boldsymbol{x},\boldsymbol{y}\in\mathcal{C}$ are non-confusable, their images $f(\boldsymbol{x}),f(\boldsymbol{y})$ are necessarily non-confusable as well and, in particular, $f(\boldsymbol{x})\neq f(\boldsymbol{y})$. To sum up, what we have just shown is that every zero-error code $\mathcal{C}\subseteq\mathcal{S}_{q}(n)$ for the $(\ell,r)$-0-insertion channel can be bijectively mapped to a subcode of $\mathcal{C}_{q,\ell,r}(n)$, implying that $|\mathcal{C}|\leq|\mathcal{C}_{q,r,\ell}(n)|$. ∎

Remark 1** (Shannon’s adjacency reducing mappings).**

The mappings having the property that the images of two non-confusable inputs are themselves non-confusable, such as the one from the preceding proof, were introduced by Shannon under the name “adjacency reducing mappings” [24]. It was shown in [24, Thm 3] that if, for a DMC with input alphabet $\mathcal{A}_{q}$, an adjacency reducing mapping $f:\mathcal{A}_{q}\to\mathcal{A}_{q}$ exists such that all letters in $f(\mathcal{A}_{q})$ are non-confusable, then the zero-error capacity of that DMC equals $\log|f(\mathcal{A}_{q})|$. This result does not directly apply in our case because the DMC to which the $(\ell,r)$-[math]-insertion channel is equivalent has an infinite alphabet, as well as costs assigned to the letters in this alphabet (see Section I-A). However, one can see from the proof of Theorem 1 that the main idea is still the same and that only several modifications to the method from [24] are needed. For example, in our setting the images $f(\boldsymbol{x})$ do not all have the same length (or “cost”), even if the originals $\boldsymbol{x}$ do. This is why we have assumed that codeword lengths are upper-bounded by $n$, as opposed to being exactly equal to $n$ (see (5) and (10)). $\blacktriangle$ **

Remark 2** (Optimal codes for the duplication channel).**

Recall from Section I-A that the $(\ell,r)$-duplication channel is equivalent to a channel which inserts up to $r$ blocks $0^{\ell}$ after every symbol of the input sequence except for the first $\ell-1$ symbols. Recall also that, in order to simplify the discussion, we have adopted the following conventions in our definition of the $(\ell,r)$-[math]-insertion channel:

(a) up to $r$ blocks $0^{\ell}$ may be inserted after every symbol of the input sequence, and

(b) the first symbol of the transmitted sequence is always non-zero.

It is clear that these assumptions have no effect on the asymptotic analysis and were adopted for convenience. In other words, dropping these assumptions would not change the zero-error capacity of the channel. However, if one wishes to study strict optimality of codes in the finite-blocklength regime, then they must be taken into account. **

We wish to point out here that optimal codes can easily be constructed by similar methods even if the above assumptions are dropped. Namely, an optimal code of length $\leq n$ for the channel which inserts up to $r$ blocks $0^{\ell}$ after every symbol of the input sequence except for the first $\ell-1$ symbols (and which has no restriction on the first symbol of a codeword) is given by

[TABLE]

where

[TABLE]

(Note that $\tilde{\mathcal{B}}_{q,\ell,r}$ is $\mathcal{B}_{q,\ell,r}$ with the first symbol of every block omitted, i.e., $\mathcal{B}_{q,\ell,r}=(\mathcal{A}_{q}\setminus\{0\})\times\tilde{\mathcal{B}}_{q,\ell,r}$.) In words, the prefix of length $\ell$ and all the non-zero symbols after that prefix are chosen arbitrarily in every codeword (as they are left intact by the channel), and the lengths of all runs of zeros after the prefix are required to be of the form (6). $\blacktriangle$ **

II-B Decoding Algorithm

Proof of Theorem 1 also outlined a simple linear-time decoding algorithm for the codes $\mathcal{C}_{q,\ell,r}(n)$, which we state here explicitly. To recover the transmitted string from the received string $\boldsymbol{z}$, the decoder first divides $\bf{z}$ into blocks of the form $\sigma\,0^{u}$, $\sigma\neq 0$, and then applies the function $f$ to each of the blocks. In other words, the decoder shortens each run of zeros in $\boldsymbol{z}$ by removing as many blocks $0^{\ell}$ as needed so that the length of the resulting run is the largest integer of the form $\frac{(ri+1)(r\ell+1)^{j}-1}{r}-1$, $1\leq i\leq\ell$, $j\geq 0$. In mathematically precise terms, the decoder’s output is

[TABLE]

where $i$ is the unique integer in $\{1,\ldots,\ell\}$ satisfying

[TABLE]

and $j$ is computed as

[TABLE]

II-C Zero-Error Capacity

Define the function

[TABLE]

In the special case $\ell=1$ this reduces to

[TABLE]

and further specializing to $q=2$, $r=1$ we get222The function $v_{2,1,1}(x)$ (in fact, the related function $x(1+v_{2,1,1}(x))=\sum_{j=0}^{\infty}x^{2^{j}}$) is of interest in transcendental number theory; see, e.g., [1].

[TABLE]

Theorem 2**.**

The zero-error capacity of the $(\ell,r)$-duplication channel is equal to $C_{0}^{\textnormal{dupl}}(q,\ell,r)=-\log\rho$, where $\rho$ is the unique positive solution to the equation $v_{q,\ell,r}(x)=1$.

Proof:

By Theorem 1 and the equivalence of the duplication and the [math]-insertion channel (see Section I-A), we conclude that the zero-error capacity of the $(\ell,r)$-duplication channel is equal to the asymptotic rate of the codes $\mathcal{C}_{q,\ell,r}(n)$, that is $C_{0}^{\textnormal{dupl}}(q,\ell,r)=\lim_{n\to\infty}\frac{1}{n}\log|\mathcal{C}_{q,\ell,r}(n)|$.

The following recurrence relation holds:

[TABLE]

which is seen by considering which of the blocks from $\mathcal{B}_{q,\ell,r}$ is the first block in each particular codeword of $\mathcal{C}_{q,\ell,r}(n)$. The theorem is then obtained by a standard application of the results of analytic combinatorics [7] after noting that the characteristic equation of the relation (LABEL:eq:Crec) is

[TABLE]

Namely, the exponential growth rate of a quantity defined by a linear recurrence with characteristic equation of the form (20) is determined by the complex root of that equation that is closest to the origin. It is known [7, Sec. IV.3] that equations of the form (20) (where the function $v_{q,\ell,r}(x)$ has only positive coefficients) have a unique positive root and that this root is necessarily closest to the origin. ∎

We note that the statements [10, Thm 16 and Cor. 18], which characterize the optimal zero-error codes and the zero-error capacity for the $(\ell,\infty)$-duplication channel, can be recovered as a special case of Theorems 1 and 2 above. To see this just observe that, as $r\to\infty$, (LABEL:eq:B) transforms to

[TABLE]

and (16) to

[TABLE]

Thus, the set of all sequences that have no run of zeros of length $\geq\ell$ ($(0,\ell-1)$-constrained sequences) is an optimal zero-error code for the $(\ell,r)$-[math]-insertion channel [10].

We also note that, in another special case ($\ell=1$; see (8) and (17)), Theorems 1 and 2 represent discrete-time analogs of the results reported in [25].

II-D Properties and Numerical Values of the Capacity

The following proposition shows how the zero-error capacity $C_{0}^{\textnormal{dupl}}(q,\ell,r)$ behaves as a function of its parameters. The statement is intuitively clear so we omit the formal proof (for example, increasing $r$ means that the channel is getting “noisier”, so the resulting capacity must be smaller).

Proposition 3**.**

The function $C_{0}^{\textnormal{dupl}}(q,\ell,r)$ has the following properties:

• (a)

For any fixed $\ell,r$, it is monotonically increasing in $q$, with \lim_{q\to\infty}\big{(}C_{0}^{\textnormal{dupl}}(q,\ell,r)-\log q\big{)}=0.

• (b)

For any fixed $q,r$, it is monotonically increasing in $\ell$, with $\lim_{\ell\to\infty}C_{0}^{\textnormal{dupl}}(q,\ell,r)=\log q$.

• (c)

For any fixed $q,\ell$, it is monotonically decreasing in $r$, with $C_{0}^{\textnormal{dupl}}(q,\ell,\infty)\coloneqq\lim_{r\to\infty}C_{0}^{\textnormal{dupl}}(q,\ell,r)=-\log\rho_{\infty}$. Here $\rho_{\infty}$ is the unique positive solution to $v_{q,\ell,\infty}(x)=1$, and $v_{q,\ell,\infty}(x)$ is defined in (22). In particular, $C_{0}^{\textnormal{dupl}}(q,1,\infty)=\log(q-1)$.

While in most cases the root $\rho$ and the capacity $-\log\rho$ are not (and most likely cannot be) given in an explicit form, their values can easily be computed/approximated numerically. Since the powers in the sum (16) grow exponentially fast in $j$, if one wishes to compute the first $d$ digits of $\rho$, then it is sufficient to approximate the infinite sum in (20) with its first $\sim\log d$ summands and find the positive root of the resulting polynomial. Moreover, this root is guaranteed to be in the range \big{[}\frac{1}{q},\frac{1}{q-1}\big{]}, which further simplifies the numerical procedures for finding it. For example, for the binary $(1,1)$-duplication channel we get $\rho\approx 0.659$ and $C_{0}^{\textnormal{dupl}}(2,1,1)=-\log\rho\approx 0.602$ bits per symbol.

As an illustration, the zero-error capacity $C_{0}^{\textnormal{dupl}}(q,\ell,r)$ is plotted in Figure 2 as a function of $r$, for several values of $q$ and $\ell$. We make here a few observations about this function based on the obtained numerical results. For most values of the parameters $q$ and $\ell$, the value of $C_{0}^{\textnormal{dupl}}(q,\ell,r)$ very quickly converges to $C_{0}^{\textnormal{dupl}}(q,\ell,\infty)$ as $r$ grows. The convergence is slowest for the binary sticky-insertion channel (for which the limiting value is $C_{0}^{\textnormal{dupl}}(2,1,\infty)=0$), and it becomes faster as either $q$ or $\ell$ (or both) increase. For example, for every $(q,\ell)$ with $q\geq 3$ and $\ell\geq 3$, the value of $C_{0}^{\textnormal{dupl}}(q,\ell,r)$ practically drops to its asymptotic value $C_{0}^{\textnormal{dupl}}(q,\ell,\infty)$ already at $r=1$. The exact value of the difference $C_{0}^{\textnormal{dupl}}(q,\ell,r)-C_{0}^{\textnormal{dupl}}(q,\ell,\infty)$, which represents the penalty incurred by assuming that an unbounded number of duplicates may be inserted at a single position (as in the model in [10]) when in fact this number is bounded by $r$, can by Theorem 2 and Proposition 3 be computed as $\log(\rho_{\infty}/\rho)$, where $\rho$ (resp. $\rho_{\infty}$) is the positive solution to $v_{q,\ell,r}(x)=1$ (resp. $v_{q,\ell,\infty}(x)=1$). Also note that, for fixed $q$ and $r$, the value of the capacity quickly converges to its trivial upper bound $\log q$ as $\ell$ grows, meaning that long duplication errors (large $\ell$) can be corrected by using codes with negligible redundancy.

III Constant-Weight Codes for the

[math]-Insertion Channel

In this section we extend the results presented above to the case where all codewords are required to have the same Hamming weight. The need to analyze constant-weight codes for the [math]-insertion channel arises naturally as this channel preserves the Hamming weight of the transmitted sequence and, therefore, an optimal code is a union of optimal constant-weight codes over all possible weights. In particular, the subcode of $\mathcal{C}_{q,\ell,r}(n)$ consisting of all codewords of weight $w$, denoted $\mathcal{C}_{q,\ell,r}(n;w)$, is an optimal zero-error code of length $n$ and weight $w$ for the $(\ell,r)$-[math]-insertion channel. This code is illustrated in Figure 3 for $q=2$, $\ell=1$, $r=1$, $n=19$, $w=2$.

Constant-weight codes are interesting in the present context for another reason – they solve quite easily the issue with codeword concatenation, i.e., consecutive transmission of multiple codewords. Namely, since the [math]-insertion channel affects the length of the transmitted sequence, if multiple codewords are being sent in succession the receiver may not be able to infer the boundaries between the output sequences that correspond to different codewords and to decode them correctly. If one uses constant-weight codes of weight $w$, however, the receiver can recognize the boundaries by simply counting the non-zero symbols and partitioning the output sequence into segments of the form $\sigma_{1}\,0^{u_{1}}\,\cdots\,\sigma_{w}\,0^{u_{w}}$, where $\sigma_{i}\neq 0$.

Remark 3**.**

We should emphasize that we are analyzing here constant-weight codes for the [math]-insertion channel, and not constant-weight codes for the duplication channel. Recalling the transformation $\phi_{\ell}$ that translates codes for one into codes for the other (see Section I-A), we see that a constant-weight requirement in the $(1,r)$-[math]-insertion channel corresponds to a constant-number-of-runs-of-identical-symbols requirement in the $(1,r)$-duplication channel. $\blacktriangle$ **

For $\omega\in[0,1]$, let C_{0}^{\textnormal{0-ins}}(\omega;q,\ell,r) denote the largest rate achievable by zero-error codes of length $n\to\infty$ and weight $w\sim\omega n$ in the $(\ell,r)$-[math]-insertion channel, i.e., the constant-weight zero-error capacity of the $(\ell,r)$-[math]-insertion channel. Let also C_{0}^{\textnormal{0-ins}}(q,\ell,r) be the zero-error capacity of the $(\ell,r)$-[math]-insertion channel (with no restriction on the weight).

Theorem 4**.**

For $\omega\in(0,1)$, the constant-weight zero-error capacity of the $(\ell,r)$-[math]-insertion channel is equal to

[TABLE]

where $\rho_{w}$ is the unique positive solution to the equation \sum_{j=0}^{\infty}\sum_{i=1}^{\ell}\Big{(}\frac{(ri+1)(r\ell+1)^{j}-1}{r}-\frac{1}{\omega}\Big{)}x^{\frac{(ri+1)(r\ell+1)^{j}-1}{r}}=0. Also, C_{0}^{\textnormal{0-ins}}(0;q,\ell,r)=0 and C_{0}^{\textnormal{0-ins}}(1;q,\ell,r)=\log(q-1).

For any fixed $q,\ell,r$, C_{0}^{\textnormal{0-ins}}(\omega;q,\ell,r) is a continuous, strictly concave function of $\omega$. It attains its maximal value at

[TABLE]

where $\rho$ is the unique positive solution to $v_{q,\ell,r}(x)=1$, andthis value is {C_{0}^{\textnormal{0-ins}}(\omega^{*};q,\ell,r)=C_{0}^{\textnormal{0-ins}}(q,\ell,r)=C_{0}^{\textnormal{dupl}}(q,\ell,r)}.

Proof:

By the observation from the first paragraph of this section we conclude that C_{0}^{\textnormal{0-ins}}(\omega;q,\ell,r)=\lim_{n\to\infty}\frac{1}{n}\log\!\big{|}\mathcal{C}_{q,\ell,r}(n;\omega n)\big{|}. Determining this limit, which represents the exponential growth rate of \big{|}\mathcal{C}_{q,\ell,r}(n;\omega n)\big{|}, is analogous to the derivation of [12, Lem. 1], which in turn is an application of the methods described in [22, Sec. 12.2], so we omit the details. The main difference with respect to [12] is that the building blocks of codewords are in the present case from $\mathcal{B}_{q,\ell,r}$, while in [12] they are from $\{1\,0^{d},\ldots,1\,0^{k}\}$. ∎

Therefore, the constant-weight codes $\mathcal{C}_{q,\ell,r}(n;\omega^{*}n)$ achieve the zero-error capacity of the $(\ell,r)$-[math]-insertion channel. For example, for $q=2$, $\ell=1$, $r=1$, we have $\rho\approx 0.659$, and the optimizing weight—the relative weight of zero-error-capacity-achieving constant-weight codes—is $\omega^{*}\approx 0.519$.

Acknowledgment

The author would like to thank the referees, for their thorough reviews and helpful comments on the original version of the manuscript, and João Ribeiro (Imperial College London), for bringing [2] to his attention.