Applications of Gaussian Binomials to Coding Theory for Deletion Error Correction

TL;DR

This paper explores the use of Gaussian binomials in coding theory, specifically for deletion error correction, by establishing new theorems on code cardinalities and deletion spheres.

Contribution

It introduces novel applications of q-binomial coefficients to analyze and determine properties of deletion-correcting codes, expanding theoretical understanding.

Findings

Cardinalities of certain error-correcting codes are determined.

A curious phenomenon related to deletion spheres is proved.

New theorems connect Gaussian binomials with deletion error correction.

Abstract

We present new applications on $q$-binomials, also known as Gaussian binomial coefficients. Our main theorems determine cardinalities of certain error-correcting codes based on Varshamov-Tenengolts codes and prove a curious phenomenon relating to deletion sphere for specific cases.

Equations

$$y_i=\{\begin{array}{ll}1& (x_i>x_{i+1}),\\ 0& (\text{otherwise}).\end{array}$$

$$\mathrm{DM}(C):=\sum _{\mathbf{x}\in C}{q}^{\rho \circ \delta (\mathbf{x})},$$

$$\rho \circ \delta (\mathbf{x})=\sum _{1\le i<n,x_i>x_{i+1}}i.$$

$$\#C=\sum _{\mathbf{x}\in C}1=\sum _{\mathbf{x}\in C}{1}^{\rho \circ \delta (\mathbf{x})}=\sum _{\mathbf{x}\in C}{1}^{\mathrm{wt}(\mathbf{x})}.$$

$$\sum _{\mathbf{x}\in C}{q}^{\rho (\mathbf{x})}{Y}^{\mathrm{wt}(\mathbf{x})},$$

$$\mathrm{DM}(C_1\cup C_2)=\mathrm{DM}(C_1)+\mathrm{DM}(C_2).$$

$$\mathrm{DM}(C_1\cup C_2)$$

$$=\sum _{\mathbf{x}\in C_1}{q}^{\rho \circ \delta (\mathbf{x})}+\sum _{\mathbf{x}\in C_2}{q}^{\rho \circ \delta (\mathbf{x})}$$

$$=\mathrm{DM}(C_1)+\mathrm{DM}(C_2)$$

$$[i]:=\frac{q^i-1}{q-1}=1+q+\cdots +q^{i-1},$$

$$[i]!$$

$$[0]!$$

$$\left[\begin{array}{c}i\\ j\end{array}\right]:=\{\begin{array}{ll}\frac{[i]!}{[j]![i-j]!}& (i\ge j\ge 0),\\ 0& (\text{otherwise}).\end{array}$$

$$\sum _{p\in L}{q}^{S(p)}=\left[\begin{array}{c}i\\ j\end{array}\right].$$

$$1+q+2q^2+q^3+q^4.$$

$$\left[\begin{array}{c}4\\ 2\end{array}\right]$$

$$=\frac{[4][3]}{[2][1]}$$

$$=\frac{(1+q+q^2+q^3)(1+q+q^2)}{(1+q)(1)}$$

$$=1+q+2q^2+q^3+q^4.$$

$$\underset{q\to \zeta }{\lim }\left[\begin{array}{c}\alpha +\beta \\ \beta \end{array}\right]=\{\begin{array}{ll}\left(\genfrac{}{}{0ex}{}{\frac{\alpha +\beta }{d}}{\frac{\beta }{d}}\right)& d|⟨\alpha ,\beta ⟩,\\ 0& \text{otherwise,}\end{array}$$

$$\underset{q\to \zeta }{\lim }\left[\begin{array}{c}cd\\ bd\end{array}\right]$$

$$\underset{q\to \zeta }{\lim }\left[\begin{array}{c}\alpha +\beta \\ \beta \end{array}\right]$$

$$=\frac{(\alpha +\beta )(\alpha +\beta -d)\dots (d)}{((\alpha )(\alpha -d)\dots (d))((\beta )(\beta -d)\dots (d))}$$

$$=\frac{(\frac{\alpha +\beta }{d})(\frac{\alpha +\beta }{d}-1)\dots (1)}{((\frac{\alpha }{d})(\frac{\alpha }{d}-1)\dots (1))((\frac{\beta }{d})(\frac{\beta }{d}-1)\dots (1))}$$

$$=\left(\genfrac{}{}{0ex}{}{\frac{\alpha +\beta }{d}}{\frac{\beta }{d}}\right).$$

$$\mathrm{DM}(C_{\alpha ,\beta })(=\sum _{0\le m<\alpha +\beta }\mathrm{DM}(C_{\alpha ,\beta ,m}))=\left[\begin{array}{c}\alpha +\beta \\ \beta \end{array}\right]$$

$$C_{\alpha ,\beta ,m}:=\{\mathbf{x}\in C_{\alpha ,\beta }\mid \rho \circ \delta (\mathbf{x})\equiv m(\mathrm{mod}\alpha +\beta )\}.$$

$$\mathrm{DM}(C_{\alpha ,\beta })=\sum _{0\le m<\alpha +\beta }\mathrm{DM}(C_{\alpha ,\beta ,m}).$$

$$\sum _{0\le m<\alpha +\beta }\#C_{\alpha ,\beta ,m}{q}^m\equiv \left[\begin{array}{c}\alpha +\beta \\ \beta \end{array}\right](\mathrm{mod}{q}^{\alpha +\beta }-1).$$

$$\left[\begin{array}{c}2+2\\ 2\end{array}\right]=1+q+2q^2+q^3+q^4.$$

$$\sum _{0\le m<4}\#C_{2,2,m}{q}^m=2+q+2q^2+q^3\equiv \left[\begin{array}{c}2+2\\ 2\end{array}\right](\mathrm{mod}{q}^4-1).$$

$$\mathrm{dS}(C)=\{ABAABB,AAAABB,AABABB,AABAAB\}.$$

$$\#\mathrm{dS}(\{\mathbf{x}\})=||\mathbf{x}||.$$

$$\#\mathrm{dS}(C)=\sum _{\mathbf{x}\in C}\#\mathrm{dS}(\{\mathbf{x}\}).$$

$$\mathrm{dS}(C)={\bigsqcup }_{\mathbf{x}\in C}\mathrm{dS}(\{\mathbf{x}\}).$$

Full text

1 Introduction

$q$-binomials [11], also known as Gaussian binomial coefficients [1] are $q$-analogs of the binomial coefficients. They are well-known and well-studied, with important combinatorial implications and have properties analogous to binomial coefficients [2, 4, 9, 13]. However, to the best of the authors’ knowledge, they have not been considered from the perspective of coding theory for deletion errors.

Terminology is defined precisely in subsequent sections, but here we give an informal description of the descent moment distribution. First, a descent vector (also studied in [7]) is a binary $01$-vector that indicates the indices of descent in an associated vector. The moment of a $01$-vector (also studied in [5, 6, 7, 14]) is a summation of the product of the index by the value of the $01$-vector. A descent moment is simply the amalgam of these two concepts, and the “descent moment distribution” of a set of vectors is the polynomial whose coefficients indicate the number of vectors having a particular descent moment in the given set.

The main contributions of this paper relate to a class of deletion codes. This provides implications for calculating the cardinality of sets that are of interest in the theory of error-correcting codes.

The descent moment distribution in the formula of the main theorem above is taken over certain sets of interest. These sets are related to a well-known class of sets studied by R. P. Stanley known as VT (Varshamov-Tenengolts) codes [12] (also known as special cases of Levenshtein codes [14]). In his study, Stanley obtained an exact formula for the cardinality of VT codes by considering a certain moment distribution in conjunction with the Hamming weight. The formula was non-trivial, and involved the sum of Möbius functions and Euler functions. His formula was for the original VT codes, but other related sets, in particular permutation and multi-permutation codes based on VT codes, do not have such formulas. Moreover, only the moment distribution was considered, not the descent moment distribution nor any relation to $q$-binomials.

Partial results were presented at the IEEE International Symposium on Information Theory (ISIT) 2018 [3].

2 Preliminaries and Remarks

2.1 Descent Moment Distribution

Let $\mathbf{x}:=x_{1}x_{2}\dots x_{l}$ be an element of $\{A,B\}^{l}$, where $\{A,B\}$ is a binary ordered set with $A<B$. Instead of $(x_{1},x_{2},\dots,x_{l})$, the notation $x_{1}x_{2}\dots x_{l}$ is used in this paper. A 01-vector $\mathbf{y}:=y_{1}y_{2}\dots y_{l-1}\in\{0,1\}^{l-1}$ is called a descent vector of $\mathbf{x}$ if

[TABLE]

We denote the descent vector of $\mathbf{x}$ by $\delta(\mathbf{x})$. Sets considered in this paper are often defined via conditions with descent vectors. For a 01-vector $\mathbf{y}$, the moment of $\mathbf{y}:=y_{1}y_{2}\dots y_{l-1}$ is defined as $y_{1}+2y_{2}+\dots+(l-1)y_{l-1}$. The moment is denoted by $\rho(\mathbf{y})$. Note that the moment does not belong to a binary field but rather it is defined as an integer, whereas $\mathbf{y}$ is a 01 vector. For a set $C$ of binary sequences, we introduce the following polynomial $\mathrm{DM}(C)$ of $q$ as our primary interest:

[TABLE]

where $\rho\circ\delta(\mathbf{x}):=\rho(\delta(\mathbf{x}))$.

Remark 2.1**.**

It is easy to see that

[TABLE]

The right hand side is well-known as the major index of $\mathbf{x}$.

In this paper, we call $\mathrm{DM}()$ the descent moment distribution for connecting coding theory, while it is the statistic of major index.

The Hamming weight distribution is an object similar to $\mathrm{DM}$ [8]. If $A=0$ and $B=1$ the distribution is defined as $\sum_{\mathbf{x}\in C}Y^{\mathrm{wt}(\mathbf{x})},$ where $\mathrm{wt}(\mathbf{x})$ is the Hamming weight of $\mathbf{x}$, i.e., the number of non-zero entries of $\mathbf{x}$. Both distributions may be applied to obtain the cardinality $\#C$ by substituting $1$ for their variable:

[TABLE]

Another related distribution with both the moment and Hamming weight is:

[TABLE]

which is used to obtain the Hamming weight distribution for VT (Varshamov-Tenengolts) codes (see 2.4 in [12]).

Notice that the descent moment distribution for the union of disjoint sets is equal to the sum of their descent moment distributions.

Lemma 2.2**.**

For $C_{1},C_{2}\subset\{A,B\}^{l}$, if $C_{1}\cap C_{2}=\emptyset$,

[TABLE]

Proof.

[TABLE]

∎

2.2 $q$-integer, $q$-factorial and $q$-binomial

The notion of a $q$-analogue is a general notion in pure mathematics for generalizing or extending a mathematical object. For a mathematical object $f$, another mathematical object $F(q)$ is called a $q$-analogue of $f$ if $f=F(1)$ or $f=\lim_{q\rightarrow 1}F(q)$.

For a positive integer $i$, the $q$-integer $[i]$ is defined as

[TABLE]

and the $q$-factorial $[i]!$ is defined as

[TABLE]

Using $q$-factorials, for non-negative integers $i$ and $j$, we define the $q$-binomial as

[TABLE]

$q$-binomial is also called a Gaussian binomial coefficient. It is easy to see that the $q$-integer $[i]$, the $q$-factorial $[i]!$, and the $q$-binomial$\begin{bmatrix}i\\ j\end{bmatrix}$ are $q$-analogues of the integer $i$, the factorial $i!$, and the binomial $\binom{i}{j}$ respectively.

$q$-binomials are known to correlate to certain weight distributions of lattice paths. Let us consider the set $L$ of lattice paths from $(0,0)$ to $(j,i-j)$. As is well-known, its cardinality $\#L$ is given by $\binom{i}{j}$. By defining the weight $S(p)$ of a path $p$ as the number of squares which are on the north-western side of the path, the following is also well-known [11] (see Example 2.3 below):

[TABLE]

Example 2.3** (Paths from $(0,0)$ to $(2,2)$).**

There are 6 paths from $(0,0)$ to $(2,2)$ (see Figure 1).

Their weights are [math], $1$, $2$, $2$, $3$, and $4$. Hence the weight distribution is

[TABLE]

On the other hand, the $q$-binomial with $i=4$ and $j=2$ is

[TABLE]

The following is used in the proof of Corollary 3.1.

Lemma 2.4**.**

Let $\zeta$ be the $d$th primitive root of $1$ and $\zeta^{\alpha+\beta}=1$.

[TABLE]

where $\langle\alpha,\beta\rangle$ is the greatest common divisor of $\alpha$ and $\beta$.

Proof.

The assumption $\zeta^{\alpha+\beta}=1$ implies that $d|\alpha+\beta$, and in particular $1\leq d\leq\alpha+\beta$. The number of zero factors of $[\alpha+\beta]!$ for substituting $\zeta$ to $q$ is $\lfloor(\alpha+\beta)/d\rfloor$ and the number of zero factors of $[\alpha]![\beta]!$ is $\lfloor\alpha/d\rfloor+\lfloor\beta/d\rfloor$.

If $d$ does not divide $\langle\alpha,\beta\rangle$, it implies $\lfloor(\alpha+\beta)/d\rfloor>\lfloor\alpha/d\rfloor+\lfloor\beta/d\rfloor$. Hence $\lim_{q\rightarrow\zeta}\begin{bmatrix}\alpha+\beta\\ \beta\end{bmatrix}=0$.

If $d$ divides $\langle\alpha,\beta\rangle$, it implies $\lfloor(\alpha+\beta)/d\rfloor=\lfloor\alpha/d\rfloor+\lfloor\beta/d\rfloor$. Note that

[TABLE]

for $0<a<d$. Hence

[TABLE]

∎

2.3 Major Index and $q$-binomial

For positive integers $\alpha$ and $\beta$, let $C_{\alpha,\beta}$ be the set of vectors with $\alpha$ entries of $A$ and $\beta$ entries of $B$. Hence $C_{\alpha,\beta}$ consists of $\binom{\alpha+\beta}{\alpha}$ elements that are obtained by all permutations to $AA\dots ABB\dots B$.

The following is well-known for major index.

Fact 2.5** (See [10]).**

For any positive integers $\alpha$ and $\beta$,

[TABLE]

We partition $C_{\alpha,\beta}$ into $\alpha+\beta$ subsets as follows:

[TABLE]

Since $\{C_{\alpha,\beta,m}\}_{0\leq m<\alpha+\beta}$ partitions $C_{\alpha,\beta}$, by Lemma 2.2, we remark the following:

Remark 2.6**.**

[TABLE]

From Fact 2.5 and the definition of $C_{\alpha,\beta,m}$, we have the following:

Corollary 2.7**.**

[TABLE]

Example 2.8** ($\alpha=\beta=2$).**

As seen in Example 2.3,

[TABLE]

On the other hand, $C_{2,2,0}=\{AABB,BABA\}$, $C_{2,2,1}=\{BAAB\}$, $C_{2,2,2}=\{ABAB,BBAA\}$, and $C_{2,2,3}=\{ABBA\}$.

Hence we verify

[TABLE]

2.4 Coding Theoretic Remarks: Deletions and Partitions via VT Codes

Deletion is a combinatorial operation for a sequence. Single deletions shorten a given sequence. For example, a sequence $AAAA$ of length $4$ changes to the sequence $AAA$ of length $3$ after a single deletion. Note that a single deletion that occurs in a string of consecutive repeated entries results in the same sequence regardless of where the deletion occurs. Indeed, the deletions in either the 1st entry or the 2nd entry from the sequence $AABAAA$ result in the same sequence $ABAAA$. Hence a sequence $AABAAA$ of length $6$ may be changed by a single deletion to one of three possible sequences of length $5$: $ABAAA$, $AAAAA$, or $AABAA$.

For a set $C$ of vectors, we define the set $\mathrm{dS}(C)$ as the set of sequences obtained by a single deletion in $C$, and call it the deletion sphere of $C$. For example, for $C:=\{AABAABB\}$,

[TABLE]

A maximal consecutive subsequence of repetitions of the same entry is called a run. For a vector $\mathbf{x}$, the number of runs is denoted by $||\mathbf{x}||$ and is called the run number in this paper. For example, $||AAAA||=1$ and $||AABAAA||=3$. The run number $||\mathbf{x}||$ is equal to the number of sequences that are obtained by single deletions to $\mathbf{x}$:

Fact 2.9** ([6]).**

For any vector $\mathbf{x}$,

[TABLE]

Hence, the cardinality of $\mathrm{dS}(C)$ for a singleton depends on its element.

A set $C$ is called a single deletion correcting code if

[TABLE]

This definition is equivalent to

[TABLE]

Levenshtein showed that the following sets $\mathrm{VT}_{l-1,m}$ are single deletion correcting codes for any positive integer $l$ and any integer $m$ [5]:

[TABLE]

This code $\mathrm{VT}_{l-1,m}$ is called a VT code. The set $C_{\alpha,\beta,m}$ is written by using VT codes:

[TABLE]

The following statement strengthens our motivation to investigate $C_{\alpha,\beta,m}$. The proof is a direct corollary of Lemma 3.2 in [7].

Theorem 2.10**.**

The set $C_{\alpha,\beta,m}$ is a single deletion correcting code.

3 Main Contributions

Our main contributions of this paper are the properties of $C_{\alpha,\beta,m}$. Theorem 3.1 and Corollary 3.2 are enumerative combinatorial results and Theorem 3.3 is a coding theoretic result.

3.1 Cardinality of $C_{\alpha,\beta,m}$

Theorem 3.1**.**

For any $\alpha,\beta$,

[TABLE]

where $\mu$ is the möbius function, $\phi$ is the Euler function, and $\langle d,m\rangle$ is the greatest common divisor of $d$ and $m$.

In particular,

[TABLE]

Proof.

Applying Eq. (2), the second half of Fact 2.5, we analyze the $q$-binomial $Q(q):=\begin{bmatrix}\alpha+\beta\\ \beta\end{bmatrix}\pmod{q^{\alpha+\beta}-1}.$ Since the polynomial $Q(q)$ is of degree at most $\alpha+\beta$, $Q(q)$ is determined by $\alpha+\beta$ different points of a complex field $\mathbb{C}$, for example the elements of the set $Z:=\{1,\zeta,\zeta^{2},\cdots,\zeta^{\alpha+\beta-1}\}$ of $\alpha+\beta$th roots of 1.

By Lemma 2.4, $Q(q)$ may be written as

[TABLE]

where $h_{t}(q)$ is a polynomial such that:

1. the degree is at most $\alpha+\beta-1$,
2. $h_{t}(z)=1$ for a primitive $t$th root of $1$,
3. $h_{t}(z)=0$ for $z\in Z$ but not a primitive $t$th root. Indeed,

[TABLE]

where $Z_{t}$ is the set of primitive $t$th roots of $1$.

Since

[TABLE]

and

[TABLE]

we have

[TABLE]

Hence by observing the coefficient of $q^{m}$,

[TABLE]

Setting $m:=0$ implies $\mu\left(\dfrac{d}{\langle d,m\rangle}\right)=1$ and $\phi\left(\frac{d}{\langle d,m\rangle}\right)=1.$ Thus we obtain the formula for $\#C_{\alpha,\beta,0}$. ∎

Corollary 3.2**.**

For any $\alpha,\beta$ and $m$,

[TABLE]

Proof.

Note that

[TABLE]

This implies $\mu(d/\langle d,m\rangle)\leq 1=\mu(d/\langle d,0\rangle)$. Similarly $\dfrac{\phi(d)}{\phi(d/\langle d,m\rangle)}\leq\phi(d)=\dfrac{\phi(d)}{\phi(d/\langle d,0\rangle))}$ holds. Therefore

[TABLE]

∎

3.2 Deletion Sphere in the Case $\alpha=\beta$

In this subsection we briefly discuss a curious phenomenon relating the cardinality of $C_{\alpha,\beta,m}$ and deletion spheres when $\alpha=\beta$.

As is mentioned in Fact 2.9, the cardinality of $\mathrm{dS}(C)$ for a singleton depends on its element. However, we have the following:

Theorem 3.3**.**

Assume $\alpha=\beta$, and set $\gamma:=\alpha(=\beta)$. Then

[TABLE]

Example 3.4** (Case $\alpha=\beta=2$).**

As we have seen in Example 2.8, $C_{2,2,0}=\{AABB,BABA\}$. Hence $\mathrm{dS}(C_{2,2,0})=||AABB||+||BABA||=2+4=3\#C_{2,2,0}$. Similarly, $\mathrm{dS}(C_{2,2,1})=||BAAB||=3=3\#C_{2,2,1}$.

Definition 3.5** ($R_{r}(q)$).**

For an integer $r$, let us define

[TABLE]

where $||\mathbf{x}||$ is the run number of $\mathbf{x}$.

Lemma 3.6**.**

[TABLE]

and

[TABLE]

Proof.

Since the run number of an element of $C_{\alpha,\beta}$ is greater than or equal to $2$ and is at most $\alpha+\beta$, Eq. (3) holds.

[TABLE]

Hence Eq. (4) holds. ∎

These two relations above will be used for the proof of Theorem 3.3. As preparation, we show the following:

Lemma 3.7**.**

[TABLE]

where $\lfloor\epsilon\rfloor$ denotes the maximal integer that does not exceed $\epsilon$.

Proof.

For the sake of brevity, we only show the case when $r$ is even. The odd case is similarly proven. Any element of $C_{\alpha,\beta}$ with $r$ runs, where $r$ is even, has one of the following two forms:

[TABLE]

[TABLE]

where $a_{j}$ ($b_{j}$) denotes the length of the $j$th run with entry A (B), and $1\leq j\leq r/2$. For case (*), the descent moment is

[TABLE]

and for case ($\ast\ast$), the descent moment is

[TABLE]

Hence

[TABLE]

where $\mathcal{A}^{*}=a_{1}+(a_{1}+a_{2})+\dots+(a_{1}+a_{2}+\dots+a_{r/2-1})$, $\mathcal{A}^{**}=\mathcal{A}^{*}$,

$\mathcal{B}^{*}=b_{1}+(b_{1}+b_{2})+\dots+(b_{1}+b_{2}+\dots+b_{r/2-1})$, and $\mathcal{B}^{**}=b_{1}+(b_{1}+b_{2})+\dots+(b_{1}+b_{2}+\dots+b_{r/2})$.

For calculating $\sum q^{\mathcal{A}^{*}}$, let us define a bijection, depicted by Figure 2, from the set of sequences $a_{1},a_{2},\dots,a_{r/2}\geq 1$ to the set of lattice paths $p$ from $(0,0)$ to $(\alpha-r/2,r/2-1)$. Note that $\mathcal{A}^{*}=S(p)+(1+2+\dots+(r/2-1))=(r/2)(r/2-1)/2+S(p)$.

Therefore by Eq. (1),

[TABLE]

Similarly

[TABLE]

Note that in the previous two equations we have used both $\lfloor(r-2)/2\rfloor$ and $\lfloor(r-1)/2\rfloor$ to represent $r/2-1$, which is permissible since $r$ is even. The choices were made so that the end result is consistent with the case when $r$ is odd. The previous two equations imply that

[TABLE]

By a similar argument, we can show

[TABLE]

Hence for even $r$,

[TABLE]

As mentioned at the beginning of the proof, the case when $r$ is odd is similarly proven. ∎

The following is the key lemma to prove Theorem 3.3. It states that a sort of symmetry of $R_{r}(q)$ on $r$ holds by the assumption $\alpha=\beta$ and considering $\pmod{q^{2\gamma}-1}$.

Lemma 3.8**.**

For $2\leq r\leq 2\gamma$,

[TABLE]

Proof.

By Lemma 3.7,

[TABLE]

$q^{\gamma(\gamma-r+1)}+q^{\gamma(\gamma-r+2)}\equiv 1+q^{\gamma}\pmod{q^{2\gamma}-1}$ holds from $\{\gamma-r+1,\gamma-r+2\}\equiv\{0,1\}\pmod{2}$. ∎

Proof of Theorem 3.3.

Define $G_{m}(q)$ as

[TABLE]

Then

[TABLE]

Hence the proof is done by showing

[TABLE]

To this end, it is enough to show

[TABLE]

Finally we have

[TABLE]

∎

4 Conclusion

In this paper we proved a relationship between descent moment distributions and $q$-binomials. To accomplish this, we employed a lattice-path approach to prove pertinent lemmas. The relationship between descent moment distributions and $q$-binomials was then applied to determine the cardinality of $C_{\alpha,\beta,m}$.

We have seen how the descent moment distribution has some interesting properties and may provide insights into other problems. Thus further investigation into descent moment distributions, especially as it relates to combinatorics, is a logical future research direction. Below we state two open questions regarding the subject.

The natural open question is to extend the main results of this paper to ternary (or more) and then arbitrary $q$-multinomials. That is, the initial part of this open question is to prove a similar relationship for the descent moment distributions of ternary subsets of $\{A,B,C\}^{l}$ with fixed multiplicities of $A$, $B$, and $C$.