Linear Programming Bounds

TL;DR

This chapter provides foundational definitions, examples, and basic facts about linear programming bounds in coding theory, aimed at upper-level students and researchers.

Contribution

It offers a concise, reference-style overview of linear programming bounds, emphasizing clarity and foundational knowledge for educational purposes.

Findings

Summarizes key concepts and definitions

Provides illustrative examples

Includes exhaustive references

Abstract

This chapter is written for the forthcoming book "A Concise Encyclopedia of Coding Theory" (CRC press), edited by W. Cary Huffman, Jon-Lark Kim, and Patrick Sol\'e. This book will collect short but foundational articles, emphasizing definitions, examples, exhaustive references, and basic facts. The target audience of the Encyclopedia is upper level undergraduates and graduate students.

Tables

Table 1. Table 1: Parameters of known codes attaining (simultaneously) the upper bound in ( 5 ) (see also [ 22 ] ) and the lower bound ( 8 ). This table appears for ( 5 ) in [ 56 ] . Here the column for the external distance s ′ superscript 𝑠 ′ s^{\prime} is added. All codes in the table are universally optimal.

$n$	$q$	$s^{\prime }\left(𝒞\right)$	$d^{\prime }\left(𝒞\right)$	Distances	$\left|𝒞\right|$	Comment
$n$	$q$	$n$	$2$	$\left\{n\right\}$	$n$	repetition code, $s^{\prime }\left(𝒞\right)=\lfloor n/2\rfloor$ for $q=2$
$n$	$q$	$n-1$	$2$	$\left\{d\right\}$	$\frac{qd}{qd-n\left(q-1\right)}$	$n>d>\frac{\left(q-1\right)n+1}{q}$ coexistence with resolvable block-designs $2-(\left|𝒞\right|,\frac{\left|𝒞\right|}{q},n-d)$ [66]
$n$	$q$	$n-2$	$3$	$\left\{\frac{\left(q-1\right)n+1}{q}\right\}$	$\left(q-1\right)n+1$	coexistence with affine resolvable block-designs $2-(\left|𝒞\right|,\frac{\left|𝒞\right|}{q},\frac{n-1}{q})$ [67]
$p^{l}q$	$q=p^m$	$n-2$	$3$	$\{n-p^l,n\}$	$nq$	$l,m=1,2,\dots$, [67]
$qh+h-q$	$q$	$n-2$	$3$	$\{n-h,n\}$	$q^3$	, [40]
$q^2+1$	$q$	$n-3$	$4$	$\{q^2-q,q^2\}$	$q^4$	ovoid in PG$(3,q)$, [20, 63]
$56$	$3$	$53$	$4$	$\{36,45\}$	$3^6$	Projective cap, Hill [50]
$78$	$4$	$75$	$4$	$\{56,64\}$	$4^6$	Projective cap, Hill [50]
$4l$	$2$	$2l-2$	$4$	$\{2l,4l\}$	$8l$	Hadamard codes
$q+2$	$q$	$q-1$	$4$	$\{q,q+2\}$	$q^3$	$2|q$, $s^{\prime }\left(𝒞\right)=2$ for $q=4$; hyperoval in PG$(2,q)$, [20]
$11$	$3$	$5$	$5$	$\{6,9\}$	$243$	projection of Golay code
$12$	$3$	$3$	$6$	$\{6,9,12\}$	$729$	Golay code [47]
$22$	$2$	$10$	$6$	$\{8,12,16\}$	$1024$	projection of Golay code
$23$	$2$	$7$	$7$	$\{8,12,16\}$	$2048$	projection of Golay code
$24$	$2$	$4$	$8$	$\{8,12,16,24\}$	$4096$	Golay code [47]
$n$	$2$	$1$	$n$	all even	$2^{n-1}$	even weight code

Equations

$$K_i^{(n,q)}(z):=\sum _{j=0}^i(-1{)}^j(q-1{)}^{i-j}{q}^j\left(\genfrac{}{}{0ex}{}{n-j}{n-i}\right)\left(\genfrac{}{}{0ex}{}{z}{j}\right),i=0,1,2,\dots ,$$

$$(i+1)K_{i+1}^{(n,q)}(z)=[i+(q-1)(n-i)-qz]K_i^{(n,q)}(z)-(q-1)(n-i+1)K_{i-1}^{(n,q)}(z)$$

$$d{\mu }_n(t):=q^{-n}\sum _{i=0}^n\left(\genfrac{}{}{0ex}{}{n}{i}\right)(q-1{)}^i\delta (t-i)dt,$$

$$⟨f,g⟩=\int f(t)g(t)d{\mu }_n(t)$$

$$\sum _{u=0}^n{K}_i^{(n,q)}(u)K_j^{(n,q)}(u)(q-1{)}^u(\genfrac{}{}{0ex}{}{n}{u})={\delta }_{i,j}{q}^n(q-1{)}^i\left(\genfrac{}{}{0ex}{}{n}{i}\right)$$

$$\sum _{u=0}^n{K}_i^{(n,q)}(u)K_u^{(n,q)}(j)={\delta }_{i,j}{q}^n.$$

$$f_i$$

$$K_i^{(n,q)}(z)K_j^{(n,q)}(z)=\sum _{u=0}^n{p}_{i,j}^u{K}_u^{(n,q)}(z)(\mathrm{mod}{K}_{n+1}^{(n,q)}(z))$$

$$T_i^{n,q}(z,w):=\sum _{j=0}^i{K}_j^{(n,q)}(z)K_j^{(n,q)}(w){\left((q-1{)}^j(\genfrac{}{}{0ex}{}{n}{j})\right)}^{-1}.$$

$$(w-z)T_i^{n,q}(z,w)=\frac{i+1}{q(q-1{)}^i(\genfrac{}{}{0ex}{}{n}{i})}\left(K_{i+1}^{(n,q)}(z)K_i^{(n,q)}(w)-K_{i+1}^{(n,q)}(w)K_i^{(n,q)}(z)\right).$$

$$B_i=\frac{1}{|\mathcal{C}|}\left|\{(x,y)\in {\mathcal{C}}^2:d(x,y)=i\}\right|,i=0,1,\dots ,n,$$

$$B_i^{\prime }=\frac{1}{|\mathcal{C}|}\sum _{j=0}^n{B}_j{K}_i^{(n,q)}(j),i=0,1,\dots ,n,$$

$$B_i^{\prime }\ge 0,i=1,2,\dots ,n.$$

$$s=|\{i:B_i\ne 0,i>0\}|,s^{\prime }=|\{i:B_i^{\prime }\ne 0,i>0\}|.$$

$$\hat{f}(z)=q^{-n/2}\sum _{j=0}^{n}f(j)K_j^{(n,q)}(z)$$

$$f(0)+\sum _{i=1}^n{B}_{i}f(i)=|\mathcal{C}|\left(f_0+\sum _{i=1}^n{f}_i{B}_i^{\prime }\right).$$

$$A_q(n,d):=\max \{|\mathcal{C}|:\mathcal{C}\subseteq F_q^n,d(\mathcal{C})=d\}.$$

$$B_q(n,\tau ):=\min \{|\mathcal{C}|:\mathcal{C}\subseteq F_q^n,\tau (\mathcal{C})=\tau \}.$$

$$\frac{f(0)}{f_0}\cdot \frac{\hat{f}(0)}{{\hat{f}}_0}=q^n.$$

$$E_h(\mathcal{C}):=\sum _{x,y\in \mathcal{C},x\ne y}h(d(x,y)).$$

$$E_h(n,M;q):=\min \{E_h(\mathcal{C}):\mathcal{C}\subseteq F_q^n,|\mathcal{C}|=M\}.$$

$$q^{d^{\prime }-1}\le |\mathcal{C}|\le q^{n-d+1}.$$

$$V_k(n,q):=\sum _{i=0}^k\left(\genfrac{}{}{0ex}{}{n}{i}\right)(q-1{)}^i,0\le k\le n,$$

$$H^{n,q}(d)=q^{\epsilon }{V}_k(n-\epsilon ,q),$$

$$H^{n,q}(d^{\prime })\le |\mathcal{C}|\le \frac{q^n}{H^{n,q}(d)}.$$

$${\xi }_k^{n-1-\epsilon ,q}+1<z\le {\xi }_{k-1+\epsilon }^{n-2+\epsilon ,q}+1.$$

$$L_k^{n,q}(z):=V_{k-1}(n,q)-(q-1{)}^k(\genfrac{}{}{0ex}{}{n}{q})\frac{K_{k-1}^{(n-1,q)}(z-1)}{K_k^{(n,q)}(z)},\text{mbox}and$$

$$L^{n,q}(z):=q^{\epsilon }{L}_k^{n-\epsilon ,q}(z),\text{mbox}if{\xi }_k^{n-1-\epsilon ,q}+1<z\le {\xi }_{k-1+\epsilon }^{n-2+\epsilon ,q}+1,$$

$$\frac{q^n}{L^{n,q}(d^{\prime })}\le |\mathcal{C}|\le L^{n,q}(d).$$

$$A_q(n,d)\le \frac{qd}{qd-(q-1)n}=L_1^{n,q}(d)$$

Full text

Linear Programming Bounds

Peter Boyvalenkov peter@math.bas.bg

Institute for Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria;

Technical Faculty, Southwestern University, Blagoevgrad, Bulgaria

Danyo Danev danyo.danev@liu.se

Department of Electrical Engineering and Department of Mathematics,

Linköping University, SE-581 83 Linköping, Sweden

1 Preliminaries – Krawtchouk polynomials, codes, and designs

Definition 1.1** (Krawtchouk polynomials).**

Let $n\geq 1$ and $q\geq 2$ be integers. The Krawtchouk polynomials are defined as

[TABLE]

where $\binom{z}{j}:=z(z-1)\cdots(z-j+1)/j!$, $z\in\mathbb{R}$.

Theorem 1.2** ([57]).**

The polynomials $K_{i}^{(n,q)}(z)$ satisfy the three-term recurrence relation

[TABLE]

with initial conditions $K_{0}^{(n,q)}(z)=1$ and $K_{1}^{(n,q)}(z)=n(q-1)-qz$.

Theorem 1.3** ([57]).**

The discrete measure

[TABLE]

where $\delta(i)$ is the Dirac-delta measure at $i\in\{0,1,\ldots,n\}$, and the form

[TABLE]

define an inner product over the class $\mathcal{P}_{n}$ of real polynomials of degree less than or equal to $n$.

Theorem 1.4** (Orthogonality relations, [57]).**

Under the inner product (2) the Krawtchouk polynomials satisfy

[TABLE]

for any $i,j\in\{0,1,\ldots,n\}$. Moreover,

[TABLE]

Theorem 1.5** (Expansion, [57]).**

If $f(z)=\sum_{i=0}^{n}f_{i}K_{i}^{(n,q)}(z)$, then

[TABLE]

Definition 1.6**.**

Define $K_{n+1}^{(n,q)}(z):=\frac{q^{n+1}}{(n+1)!}\prod_{u=0}^{n}(u-z)$. Note that $K_{n+1}^{(n,q)}(z)$ is orthogonal to any polynomial $K_{i}^{(n,q)}(z)$, $i=0,1,\ldots,n$, with respect to the measure (1).

Theorem 1.7** (Krein condition, [57]).**

For any $i,j\in\{0,1,\ldots,n\}$

[TABLE]

with $p_{i,j}^{u}=0$ if $i+j>u$, $p_{i,j}^{u}>0$ if $i+j=u\leq n$, and $p_{i,j}^{u}\geq 0$ otherwise.

Definition 1.8**.**

Denote

[TABLE]

Theorem 1.9** (Christoffel-Darboux formula, [57]).**

*For any

$i=0,1,\ldots,n$ and any real $z$ and $w$*

[TABLE]

Let $\mathcal{C}\subseteq F_{q}^{n}$ be a code, where $F_{q}=\{0,1,\ldots,q-1\}$ is the alphabet of $q$ symbols (so $q$ is not necessarily a power of a prime). For $x,y\in F_{q}^{n}$, recall that $d(x,y)$ is the number of coordinates where $x$ and $y$ disagree.

Definition 1.10**.**

The vector $B(\mathcal{C})=(B_{0},B_{1},\ldots,B_{n})$, where

[TABLE]

is called the distance distribution of $\mathcal{C}$. Clearly, $B_{0}=1$ and $B_{i}=0$ for $i=1,2,\ldots,d-1$, where $d$ is the minimum distance of $\mathcal{C}$.

Definition 1.11**.**

The vector $B^{\prime}(\mathcal{C})=(B^{\prime}_{0},B^{\prime}_{1},\ldots,B^{\prime}_{n})$, where

[TABLE]

is called the dual distance distribution of $\mathcal{C}$ or the MacWilliams transform of $B(\mathcal{C})$. Obviously $B^{\prime}_{0}=1$.

Theorem 1.12** ([36, 37]).**

The dual distance distribution of $\mathcal{C}$ satisfies

[TABLE]

Theorem 1.13** ([60]).**

If $q$ is a power of a prime and $\mathcal{C}$ is a linear code in $\mathbb{F}_{q}^{n}$, then $B^{\prime}(\mathcal{C})$ is the distance distribution of the dual code $\mathcal{C}^{\perp}$.

Definition 1.14**.**

The smallest positive integer $i$ such that $B^{\prime}_{i}\neq 0$ is called the dual distance of $\mathcal{C}$ and is denoted by $d^{\prime}=d^{\prime}(\mathcal{C})$. Denote by $s=s(\mathcal{C})$ (resp., $s^{\prime}=s^{\prime}(\mathcal{C})$) the number of nonzero $B_{i}$’s (resp., $B^{\prime}_{i}$’s), $i\in\{1,2,\ldots,n\}$, i.e.

[TABLE]

The number $s^{\prime}$ is called the external distance of $\mathcal{C}$. Define $\delta=0$ if $B_{n}=0$ and $\delta=1$ otherwise (respectively, $\delta^{\prime}=0$ if $B^{\prime}_{n}=0$ and $\delta^{\prime}=1$ otherwise).

Definition 1.15**.**

Let $\mathcal{C}\subseteq F_{q}^{n}$ be a code and $M$ be a codeword matrix consisting of all vectors of $\mathcal{C}$ as rows. Then $\mathcal{C}$ is called a $\tau$-design if any set of $\tau$-columns of $M$ contains any $\tau$-tuple of $F_{q}^{\tau}$ the same number of times (namely, $\lambda:=|\mathcal{C}|/q^{\tau}$). The largest positive integer $\tau$ such that $\mathcal{C}$ is a $\tau$-design is called the strength of $\mathcal{C}$ and is denoted by $\tau(\mathcal{C})$. The number $\lambda$ is called the index of $\mathcal{C}$.

Remark 1.16**.**

A $\tau$-design in $F_{q}^{n}$ is also called an orthogonal array of strength $\tau$ [37] or a $\tau$-wise independent set [4] (see also [49]).

Theorem 1.17** ([36, 37]).**

If $\mathcal{C}\subseteq F_{q}^{n}$ has dual distance $d^{\prime}=d^{\prime}(\mathcal{C})$, then $\mathcal{C}$ is a $\tau$-design, where $\tau=d^{\prime}-1$.

Definition 1.18**.**

For a real polynomial $f(z)=\sum_{i=0}^{n}f_{i}K_{i}^{(n,q)}(z)$, the polynomial

[TABLE]

is called the dual to $f(z)$. Note that the dual to $\widehat{f}(z)$ is $f(z)$ and also that $\widehat{f}(i)=q^{n/2}f_{i}$ for any $i\in\{0,1,\dots,n\}$.

2 General linear programming theorems

Theorem 2.1** ([36, 37]).**

For any code $\mathcal{C}\subseteq F_{q}^{n}$ with distance distribution $(B_{0},B_{1},\ldots,B_{n})$ and dual distance distribution $(B^{\prime}_{0},B^{\prime}_{1},\ldots,B^{\prime}_{n})$, and any real polynomial $f(z)=\sum_{i=0}^{n}f_{i}K_{i}^{(n,q)}(z)$, it is valid that

[TABLE]

In Definition 1.9.1 in Chapter 1, $A_{q}(n,d)$ is defined for codes over $\mathbb{F}_{q}$. We now extend that definition to codes over $F_{q}$ as only the alphabet size is important and not its structure.

Definition 2.2**.**

For fixed $q$, $n$, and $d\in\{1,2,\ldots,n\}$ denote

[TABLE]

Definition 2.3**.**

For fixed $q$, $n$, and $\tau\in\{1,2,\ldots,n\}$ denote

[TABLE]

Theorem 2.4** (Linear Programming Bound for codes [36, 37]).**

Let the real polynomial $f(z)=\sum_{i=0}^{n}f_{i}K_{i}^{(n,q)}(z)$ satisfy the conditions

• (A1)

$f_{0}>0$, $f_{i}\geq 0$ for $i=1,2,\ldots,n$;

• (A2)

$f(0)>0$, $f(i)\leq 0$ for $i=d,d+1,\ldots,n$.

Then $A_{q}(n,d)\leq f(0)/f_{0}$. Equality holds for codes $\mathcal{C}\subseteq F_{q}^{n}$ with $d(\mathcal{C})=d$, distance distribution $(B_{0},B_{1},\ldots,B_{n})$, dual distance distribution $(B^{\prime}_{0},B^{\prime}_{1},\ldots,B^{\prime}_{n})$ and polynomials $f(z)$ such that $B_{i}f(i)=0$ and $B^{\prime}_{i}f_{i}=0$ for every $i=1,2,\ldots,n$.

Remark 2.5**.**

The polynomial $f$ from Theorem 2.4 is implicit in Theorem 1.9.23 a) from Chapter 1 as $\sum_{i=0}^{n}B_{i}K_{i}^{(n,q)}(i)$ and it is normalized for $f_{0}$ (or $B_{0}$) to be equal to 1; note also the normalization of the Krawtchouk polynomials. Thus the bound $f(1)/f_{0}$ appears as $\sum_{i=0}^{n}B_{i}$ in Chapter 1.

Theorem 2.6** (Linear Programming Bound for designs [36, 37]).**

Let the real polynomial $f(z)=\sum_{i=0}^{n}f_{i}K_{i}^{(n,q)}(z)$ satisfy the conditions

• (B1)

$f_{0}>0$, $f_{i}\leq 0$ for $i=\tau+1,\tau+2,\ldots,n$;

• (B2)

$f(0)>0$, $f(i)\geq 0$ for $i=1,2,\ldots,n$.

Then $B_{q}(n,\tau)\geq f(0)/f_{0}$. Equality holds for designs $\mathcal{C}\subseteq F_{q}^{n}$ with $\tau(\mathcal{C})=\tau$, distance distribution $(B_{0},B_{1},\ldots,B_{n})$, dual distance distribution $(B^{\prime}_{0},B^{\prime}_{1},\ldots,B^{\prime}_{n})$ and polynomial $f(z)$ such that $B_{i}f(i)=0$ and $B^{\prime}_{i}f_{i}=0$ for every $i=1,2,\ldots,n$.

Remark 2.7**.**

The Rao Bound in Theorem 3.3 and Levenshtein Bound in 3.10 below are obtained by suitable polynomials in Theorems 2.4 and 2.6, respectively. Examples of codes attaining these bounds are listed in Table 1 below.

Theorem 2.8** (Duality, [59]).**

A real polynomial $f(z)$ satisfies the conditions (A1) and (A2) if and only if its dual polynomial $\widehat{f}$ satisfies the conditions (B1) and (B2). Moreover,

[TABLE]

Remark 2.9**.**

Rephrased, the duality means that for any polynomial $f$ which is good for linear programming for codes, its dual is good for linear programming for designs (and conversely). Thus we obtain, in a sense, bounds for free. In particular, the duality justifies the pairs of bounds in Theorems 3.1, 3.3 and 3.10 below.

See also [29, 39, 45, 46, 59].

Definition 2.10**.**

For a code $\mathcal{C}\subseteq F_{q}^{n}$ and a function $h:\{1,2,\ldots,n\}\to\mathbb{R}$ the potential energy of $\mathcal{C}$ with respect to $h$ is defined to be

[TABLE]

Definition 2.11**.**

For fixed $q$, $n$, $h$, and $M\in\{2,3,\ldots,q^{n}\}$ denote

[TABLE]

Theorem 2.12** (Linear Programming Bound for energy of codes [34]).**

Let $n$ and $q$ be fixed, $h:(0,n]\to(0,+\infty)$ be a function, and $M\in\{2,3,\ldots,q^{n}\}$. Let the real polynomial $f(z)=\sum_{i=0}^{n}f_{i}K_{i}^{(n,q)}(z)$ satisfy the conditions

• (D1)

$f_{0}>0$, $f_{i}\geq 0$ for $i=1,2,\ldots,n$;

• (D2)

$f(0)>0$, $f(i)\leq h(i)$ for $i=1,2,\ldots,n$.

Then $E_{h}(n,M;q)\geq M(f_{0}M-f(0))$. Equality holds for codes $\mathcal{C}\subseteq F_{q}^{n}$ with distance distribution $(B_{0},B_{1},\ldots,B_{n})$, dual distance distribution $(B^{\prime}_{0},B^{\prime}_{1},\ldots,B^{\prime}_{n})$ and polynomials $f(z)$, such that $B_{i}\left[f(i)-h(i)\right]=0$ and $B^{\prime}_{i}f_{i}=0$ for every $i=1,2,\ldots,n$.

Remark 2.13**.**

Theorems 2.4, 2.6 and 2.12 can be applied with the usual simplex method for quite large parameters. For instance, the website [65] (Delsarte, a.k.a. Linear Programming (LP), upper bounds) offers a tool for computation of bounds via integer LP with Theorem 2.4. Several websites maintain tables of best known bounds (lower and upper) for codes of relatively small lengths (see, e.g. [28]).

3 Universal bounds

The Singleton Bound presented in Theorem 1.9.10 in Chapter 1 is an upper bound on the code cardinality, given $q$, $n$, and $d$. It is the upper bound in (3) below. Its proof by linear programming and the duality imply the lower bound in (3).

Theorem 3.1** (Singleton Bound [69]).**

For any code $\mathcal{C}\subseteq F_{q}^{n}$ with minimum distance $d$ and dual distance $d^{\prime}$

[TABLE]

The bounds (3) can be attained only simultaneously and this happens if and only if $d+d^{\prime}=n+2$ and all possible distances are realized (so the attaining code is an MDS code).

Definition 3.2**.**

For fixed $q$, $n$, and $d$ denote by

[TABLE]

(volume of a sphere of radius $k$ in $F_{q}^{n}$) and

[TABLE]

where $d=2k+1+\varepsilon$, $\varepsilon\in\{0,1\}$.

The Sphere Packing or Hamming Bound presented in Theorem 1.9.6 in Chapter 1 is another upper bound on the code size, given $q$, $n$, and $d$. Rao gave a lower bound on the code size, given $q$, $n$, and $d^{\prime}$. These bounds are combined in the following theorem, as they are connected by the duality.

Theorem 3.3** **(Rao Bound [64] and Hamming (sphere packing) Bound

[48]).

For any code $\mathcal{C}\subseteq F_{q}^{n}$ with minimum distance $d$ and dual distance $d^{\prime}$

[TABLE]

Definition 3.4**.**

Codes attaining the upper bound in (4) are called perfect. Designs attaining the lower bound in (4) are called tight.

Recall the definitions of $s$, $s^{\prime}$, $\delta$, and $\delta^{\prime}$ in Definition 1.14.

Theorem 3.5**.**

• a)

A code $\mathcal{C}\subseteq F_{q}^{n}$ ($|\mathcal{C}|>1)$* is a tight design if and only if $d^{\prime}=2s-\delta+1$.*

• b)

A code $\mathcal{C}\subseteq F_{q}^{n}$ ($|\mathcal{C}|>1)$* is a perfect code if and only if $d=2s^{\prime}-\delta^{\prime}+1$.*

See also [12, 42, 59, 64].

Definition 3.6**.**

For fixed $n$, $q$, and $i$ denote by $\xi_{i}^{n,q}$ the smallest root of the Krawtchouk polynomial $K_{i}^{(n,q)}(z)$. In the case of $i=0$ set $\xi_{0}^{n,q}=n+1$.

Lemma 3.7** ([57]).**

The following are valid:

• a)

For $i=1,2,\ldots,n$, $\xi_{i}^{n-2,q}<\xi_{i}^{n-1,q}<\xi_{i-1}^{n-2,q}$.

• b)

For $z\in[1,n]$, there exists a unique $k=k(z)\in\{1,2,\ldots,n\}$ and a unique $\varepsilon=\varepsilon(z)\in\{0,1\}$ such that

[TABLE]

Example 3.8**.**

One has $\xi_{0}^{n,q}=n+1$, $\xi_{1}^{n,q}=\frac{(q-1)n}{q}$ and $\xi_{2}^{n,q}=\frac{2(q-1)n-q+2-\sqrt{4(q-1)n+(q-2)^{2}}}{2q}$.

Definition 3.9**.**

For $z\in[1,n]$, let

[TABLE]

[TABLE]

where $k=k(z)$ and $\varepsilon=\varepsilon(z)\in\{0,1\}$ are as in Lemma 3.7b).

Theorem 3.10** (Levenshtein Bound [55, 57, 58]).**

For any code $\mathcal{C}\subseteq F_{q}^{n}$ with minimum distance $d$ and dual distance $d^{\prime}$

[TABLE]

The lower (upper) bound is attained if and only if $d\geq\max\{2s^{\prime}-\delta^{\prime},2\}$ (respectively, $d^{\prime}\geq\max\{2s-\delta,2\}$).

Example 3.11**.**

In the first three relevant intervals, the Levenshtein (upper) Bound is given by

[TABLE]

(which is the Plotkin Bound; discussed in Section 1.9.3) if $n-\frac{n-1}{q}=\xi_{1}^{n-1,q}+1\leq d\leq\xi_{0}^{n-2,q}+1=n$,

[TABLE]

if $\xi_{1}^{n-2,q}+1\leq d\leq\xi_{1}^{n-1,q}+1$, and

[TABLE]

if $\xi_{2}^{n-1,q}+1\leq d\leq\xi_{1}^{n-2,q}+1$.

Remark 3.12**.**

It is also worth noting two important values of the Levenshtein Bound:

[TABLE]

i.e. the Levenshtein Bound at the ends of the intervals $\left[\xi_{k}^{n-1-\varepsilon,q}+1,\xi_{k-1+\varepsilon}^{n-1,q}+1\right]$ coincides with corresponding Rao Bound.

Recall the kernels $T_{i}^{n,q}$ from Definition 1.8 and the parameters $k=k(d)$ and $\varepsilon=\varepsilon(d)$ from Lemma 3.7 b). The next theorem gives a Gauss-Jacobi quadrature formula (6), introduced by Levenshtein in [56, 57], which is instrumental in proofs of Theorem 3.16 and Theorem 3.21 a). Theorem 3.13 also introduces parameters needed for the universal bound (8) below.

Theorem 3.13** ([57]).**

For any $d\in\{1,2,\ldots,n\}$ the polynomial

[TABLE]

where $k=k(d)$ and $\varepsilon=\varepsilon(d)\in\{0,1\}$, has $k+1+\varepsilon$ simple zeros

[TABLE]

with $\alpha_{k+\varepsilon}=n$ if and only if $\varepsilon=1$ or $\varepsilon=0$ and $d=\xi_{k-1}^{n-2,r}+1$.

Moreover, for any real polynomial $f(z)$ of degree at most $2k-1+\varepsilon$ the following equality holds:

[TABLE]

where all coefficients (weights) $\rho_{i}^{(d)}$, $i=1,2,\ldots,k$, are positive, and, in the case $\varepsilon=1$, $\rho_{k+1}^{(d)}\geq 0$ with equality if and only if $d=\xi_{k}^{n-1,q}+1$. We have

[TABLE]

$i=1,2,\ldots,k$, and

[TABLE]

Remark 3.14**.**

Levenshtein used the polynomial

[TABLE]

to obtain the bounds (5). It was shown in [25] that its zeros $\alpha_{1},\ldots,\alpha_{k+\varepsilon}$ strongly suggest the optimal choice of nodes for the simplex method of Theorem 2.4 (equivalently, Theorem 1.9.23 a) from Chapter 1). Computational experiments show that simple replacement of any double zero $\alpha_{i}\in(j,j+1)$ of Levenshtein’s polynomial (7) by two simple zeros $j$ and $j+1$ gives in most cases (conjecture: for every sufficiently large rate $d/n$) the best result that can ever be obtained from Theorem 2.4.

Remark 3.15**.**

The assertions of Theorem 3.13 remain true when $d\in[1,n]$ is a continuous variable. In particular, the quadrature formula (6) holds true for any real polynomial $f(z)$ of degree at most $2k-1+\varepsilon$ with well defined roots of $g_{d}(z)$ and corresponding positive $\rho_{i}^{(d)}$, $i=1,2,\ldots,k+1$.

Theorem 3.16** ([68, 56]).**

The upper bound in (5) cannot be improved by a real polynomial $f$ of degree at most $2k-1+\varepsilon$ satisfying (A1) and the condition $f(z)\leq 0$ for $z\in[d,n]$.

Definition 3.17** (Test functions [23]).**

For $j\in\{1,2,\ldots,n\}$ and $d\in(0,n]$ denote by

[TABLE]

where $k=k(d)$, $\varepsilon=\varepsilon(d)$, and $\rho_{i}^{(d)}$ are as in Theorem 3.13. Note that $P_{j}^{n,q}=0$ for $j\leq 2k-1+\varepsilon$. Note also that $K_{j}^{(n,q)}(0)=(q-1)^{j}\binom{n}{j}$.

Theorem 3.18** ([23, 58]).**

The upper bound in (5) can be improved by a real polynomial $f$ of degree at least $2k+\varepsilon$ satisfying (A1) and the condition $f(z)\leq 0$ for $z\in[d,n]$ if and only if $P_{j}^{n,q}(d)<0$ for some $j\geq 2k+\varepsilon$.

Definition 3.19**.**

A function $h:(0,n]\to(0,+\infty)$ is called (strictly) completely monotone if $(-1)^{i}h^{(i)}(z)\geq 0$ ($>0$) for every nonnegative integer $i$ and every $z\in(0,n]$. The derivatives $h^{(i)}(z)$ can be discrete (then $z\in\{1,2,\ldots,n\}$) or continuous.

Theorem 3.20** (Universal lower bound on energy [27]).**

Let $n$, $q$ and $M\in\{2,3,\ldots,q^{n}\}$ be fixed and $h:(0,n]\to(0,+\infty)$ be a completely monotone function. If $M\in\left(H^{n,q}(2k-1+\varepsilon),H^{n,q}(2k+\varepsilon)\right]$ and $d=d(M)\in[1,n]$ is the smallest real root of the equation $M=L_{2k-1+\varepsilon}^{n,q}(d)$, then

[TABLE]

where the parameters $\alpha_{i}$ and $\rho_{i}^{(d)}$ are determined as in Theorem 3.13. Equality is attained if and only if there exists a code with $M$ codewords which attains the upper bound in (5).

Theorem 3.21** ([27]).**

Let $h$ be a strictly absolutely monotone function. The bound (8):

• a)

cannot be improved by a real polynomial $f$ of degree at most $2k-1+\varepsilon$ satisfying (A1) and $f(z)\leq h(z)$ for $z\in[d,n]$;

• b)

can be improved by a real polynomial $f$ of degree at least $2k+\varepsilon$ satisfying (A1) and the condition $f(z)\leq h(z)$ for $z\in[d,n]$ if and only if $P_{j}^{n,r}(d)<0$ for some $j\geq 2k+\varepsilon$.

Definition 3.22**.**

A code $\mathcal{C}\subseteq F_{q}^{n}$ is called universally optimal if it (weakly) minimizes potential energy among all configurations of $|\mathcal{C}|$ codewords in $F_{q}^{n}$ for each completely monotone function $h$.

The conditions for attaining the bounds (5) and (8) coincide. Thus, any code which attains the upper bound in (5) for its cardinality (and therefore the lower bound for (8) for its energy) is universally optimal. We summarize this in the next theorem.

Theorem 3.23** ([34, 27]).**

All codes which attain the upper bound in (5) are universally optimal.

Theorem 3.24** ([34]).**

Let $\mathcal{C}\subseteq F_{q}^{n}$ be a code and $h:(0,1,\ldots,n]\to\mathbb{R}$ be any function such that the bound in Theorem 2.12 is attained. Let $c\in\mathcal{C}$. Then

[TABLE]

In particular, if $\mathcal{C}$ is proved to be universally optimal by Theorem 2.12, then $\mathcal{C}\setminus\{c\}$ is universally optimal for all $c\in\mathcal{C}$ as well.

Example 3.25**.**

The Kerdock codes $\mathcal{K}_{\ell}\subset\mathbb{F}_{2}^{2^{2\ell}}$ [52] are nonlinear codes existing for lengths $n=2^{2\ell}$. Their cardinality is $|\mathcal{K}_{\ell}|=n^{2}=2^{4\ell}$ and their distance (weight) distribution is as follows:

[TABLE]

[TABLE]

It is easy to check that the Kerdock codes are asymptotically optimal with respect to the upper bound in (5) and the bound (8) as they are very close to the bounds already for small $\ell$.

Definition 3.26**.**

For any code $\mathcal{C}\subseteq F_{q}^{n}$, $\rho(\mathcal{C}):=\max\{d(x,\mathcal{C}):x\in F_{q}^{n}\}$ is called the covering radius of $\mathcal{C}$. Here $d(x,\mathcal{C}):=\min\{d(x,c):c\in\mathcal{C}\}$.

Theorem 3.27** ([37], Delsarte Bound).**

For any code $\mathcal{C}\subseteq F_{q}^{n}$ with external distance $s^{\prime}$

[TABLE]

Theorem 3.28** ([73, 74], Tietäväinen Bound).**

For any code $\mathcal{C}\subseteq F_{q}^{n}$ with dual distance $d^{\prime}=2k-1+\varepsilon$, where $k$ is positive integer and $\varepsilon\in\{0,1\}$,

[TABLE]

See also [44, 70].

Asymptotic versions of some of the bounds in this section can be found in Section 1.9.8 of Chapter 1 (see [1, 2, 13, 58, 61]).

4 Linear programming on $\mathbb{S}^{n-1}$

Definition 4.1**.**

Let $\mathbb{S}^{n-1}=\{{\bf x}=(x_{1},x_{2},\dots,x_{n}):x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1\}$ be the unit sphere in $\mathbb{R}^{n}$. The Euclidean distance between ${\bf x}=(x_{1},x_{2},\dots,x_{n})$ and ${\bf y}=(y_{1},y_{2},\dots,y_{n})$ is

[TABLE]

The inner product is defined as

[TABLE]

Note that on $\mathbb{S}^{n-1}$ the distance and the inner product are connected by

[TABLE]

Definition 4.2**.**

An $(n,M,s)$-spherical code is a non-empty finite set $\mathcal{C}\subset\mathbb{S}^{n-1}$ with cardinality $|\mathcal{C}|=M$ and maximal inner product

[TABLE]

The minimum distance $d=d(C):=\min\{d({\bf x},{\bf y}):{\bf x},{\bf y}\in\mathcal{C},{\bf x}\neq{\bf y}\}$ and the maximal inner product are connected by

[TABLE]

Definition 4.3**.**

For fixed $n$ and $s$ denote by

[TABLE]

Definition 4.4**.**

For $a,b\in\{0,1\}$ denote by $\{P_{i}^{a,b}(t)\}_{i=0}^{\infty}$ the Jacobi polynomials $\{P_{i}^{\alpha,\beta}(t)\}_{i=0}^{\infty}$ ([3, Chapter 22], [72]) with

[TABLE]

normalized by $P_{i}^{\alpha,\beta}(1)=1$. When $(a,b)=(0,0)$ we get the Gegenbauer polynomials and use the $(n)$ indexing instead of $0,0$. Denote by $t_{i}^{a,b}$ the greatest zero of the polynomial $P_{i}^{a,b}(t)$ and define $t_{0}^{1,1}=-1$. Let

[TABLE]

Theorem 4.5** ([3, 72]).**

The Gegenbauer polynomials $\{P_{i}^{(n)}(t)\}_{i=0}^{\infty}$ satisfy the recurrence relations

[TABLE]

where $P_{0}^{(n)}(t)=1$ and $P_{1}^{(n)}(t)=t$.

Theorem 4.6** ([3, 72]).**

The Gegenbauer polynomials are orthogonal on $[-1,1]$ with respect to the measure

[TABLE]

where $\gamma_{n}:=\Gamma(\frac{n}{2})/\sqrt{\pi}\Gamma(\frac{n-1}{2})$ is a normalizing constant.

Theorem 4.7** (Linear Programming Bound for spherical codes, [38, 51]).**

Let $n\geq 2$ and $f(t)$ be a real polynomial such that

• (A1)

$f(t)\leq 0$* for $-1\leq t\leq s$;*

• (A2)

The coefficients in the Gegenbauer expansion $f(t)=\sum_{i=0}^{\deg(f)}f_{i}P_{i}^{(n)}(t)$ satisfy $f_{0}>0$, $f_{i}\geq 0$ for $i=1,\ldots,\deg(f)$.

Then $A(n,s)\leq f(1)/f_{0}$.

Theorem 4.8** (Levenstein Bound [54, 56]).**

For the quantity $A(n,s)$ we have

[TABLE]

where $I_{\tau}$ is the interval

[TABLE]

Example 4.9**.**

The first three bounds in (9) are $A(n,s)\leq(s-1)/s$ for $s\in[-1,-1/n]$,

[TABLE]

for $s\in[-1/n,0]$ and

[TABLE]

for $s\in\left[0,\frac{\sqrt{n+3}-1}{n+2}\right]$.

See also [43, 55, 58].

Definition 4.10**.**

A spherical code $\mathcal{C}\subset\mathbb{S}^{n-1}$ is a spherical $\tau$-design if and only if

[TABLE]

($\sigma_{n}$ is the normalized $(n-1)$-dimensional Hausdorff measure) holds for all polynomials $p({\bf x})=p(x_{1},x_{2},\ldots,x_{n})$ of degree at most $\tau$.

Theorem 4.11** (Linear Programming Bound for spherical designs, [38]).**

Let $n\geq 2$, $\tau\geq 1$ and $f(t)$ be a real polynomial such that

• (B1)

$f(t)\geq 0$* for $-1\leq t\leq 1$;*

• (B2)

the coefficients in the Gegenbauer expansion $f(t)=\sum_{i=0}^{\deg(f)}f_{i}P_{i}^{(n)}(t)$ satisfy $f_{0}>0$, $f_{i}\leq 0$ for $i=\tau+1,\ldots,\deg(f)$.

Then any spherical $\tau$-design $\mathcal{C}\subset\mathbb{S}^{n-1}$ has cardinality $|\mathcal{C}|\geq f(1)/f_{0}$.

Theorem 4.12** (Delsarte-Goethals-Seidel Bound [38]).**

Any $\tau$-design $\mathcal{C}\subseteq\mathbb{S}^{n-1}$ has cardinality

[TABLE]

where $\tau=2k-1+\varepsilon$, $\varepsilon\in\{0,1\}$.

Definition 4.13**.**

A spherical $\tau$-design on $\mathbb{S}^{n-1}$ is called tight if it attains the bound (10).

Theorem 4.14** ([9, 10]).**

Let $n\geq 3$. Tight spherical $\tau$-designs on $\mathbb{S}^{n-1}$ exist for $\tau=1$, $2$ and $3$ for every $n\geq 2$, and possibly for $\tau=4$, $5$, $7$, and $11$. Tight spherical $4$-designs on $\mathbb{S}^{n-1}$ exist for $n=6$, $22$, and possibly for $m^{2}-3$, where $m\geq 7$ is an odd integer. Tight spherical $5$-designs on $\mathbb{S}^{n-1}$ exist for $n=3$, $7$, $23$, and possibly for $n=m^{2}-2$, where $m\geq 7$ is an odd integer. Tight spherical $7$-designs on $\mathbb{S}^{n-1}$ exist for $n=8$, $23$, and possibly for $n=3m^{2}-4$, where $m\geq 4$ is an integer. Tight spherical $11$-designs on $\mathbb{S}^{n-1}$ exist only for $n=24$.

Remark 4.15** ([38]).**

Tight spherical 4- and 5-designs coexist and are known for $m=3$ and $5$ only. Tight spherical 7-designs are known for $m=2$ and $3$ only.

See also [11, 12, 18] for general theory and [21, 44, 71] for other bounds for designs.

Theorem 4.16** ([54, 55, 56, 58]).**

The bounds and (9) and (10) are related by the equalities

[TABLE]

at the ends of the intervals $I_{\tau}$ ($\tau=2k-1+\varepsilon$, $\varepsilon\in\{0,1\}$). In particular, if $\mathcal{C}\subseteq\mathbb{S}^{n-1}$ is a tight spherical $\tau$-design, then it attains (9) in the left end of the interval $I_{\tau}$.

Definition 4.17**.**

Given an (extended real-valued) function $h(t):[-1,1]\to[0,+\infty]$, the $h$-energy (or potential energy) of $\mathcal{C}$ is given by

[TABLE]

Definition 4.18**.**

For fixed $n$, $h$, and $M\geq 2$ denote by

[TABLE]

Theorem 4.19** (Linear Programming Bound for energy [75]).**

Let $n\geq 2$, $M\geq 2$, $h:[-1,1]\to[0,+\infty]$, and $f(t)$ be a real polynomial such that

• (C1)

$f(t)\leq h(t)$* for $t\in[-1,1]$, and*

• (C2)

the coefficients in the Gegenbauer expansion $f(t)=\sum_{i=0}^{\deg(f)}f_{i}P_{i}^{(n)}(t)$ satisfy $f_{0}>0$, $f_{i}\geq 0$ for $i=1,\ldots,\deg(f)$.

Then

[TABLE]

Definition 4.20**.**

A real valued extended function $h(t):[-1,1]\to(0,+\infty]$ is called absolutely monotone if $h^{(k)}(t)\geq 0$, for every $t\in[-1,1)$ and every integer $k\geq 0$, and $h(1)=\lim_{t\to 1^{-}}h(t)$.

The following absolutely monotone potential functions are commonly used: Newton potential

[TABLE]

Riesz $s$-potential

[TABLE]

and Gaussian potential

[TABLE]

Definition 4.21**.**

For $(a,b)=(0,0),(1,0),$ and $(1,1)$ set

[TABLE]

where $r_{i}^{a,b}=1/c^{a,b}\int_{-1}^{1}\left(P_{i}^{a,b}(t)\right)^{2}(1-t)^{a}(1+t)^{b}d\mu(t)$, $c^{1,0}=c^{0,0}=\gamma_{n}$, and $c^{1,1}=\gamma_{n+2}$ (see Theorem 4.6 for relevant notation). Note that $r_{i}^{0,0}=r_{i}$.

Let $\alpha_{1}<\alpha_{2}<\cdots<\alpha_{k+\varepsilon}$ be the roots of the polynomial used for obtaining the Levenshtein Bound $L_{\tau}(n,s)$ with $s=\alpha_{k+\varepsilon}$, $\tau=2k-1+\varepsilon$, $\varepsilon\in\{0,1\}$, and let

[TABLE]

[TABLE]

Theorem 4.22** ([26]).**

Let $n\geq 2$, $\tau=2k-1+\varepsilon\geq 1$, $\varepsilon\in\{0,1\}$, and $h$ be absolutely monotone in $[-1,1]$. For $M\in\left(D(n,\tau),D(n,\tau+1)\right]$ let $s$ be the largest root of the equation $M=L_{\tau}(n,s)$ and $\alpha_{1}<\alpha_{2}<\cdots<\alpha_{k+\varepsilon}=s$, $\rho_{1},\rho_{2},\ldots,\rho_{k+\varepsilon}$ be as in Definition 4.21. Then

[TABLE]

If an $(n,M,s)$ code attains (12), then it is a spherical $\tau$-design and its inner products form the set $\{\alpha_{1},\alpha_{2},\ldots,\alpha_{k+\varepsilon}\}$.

See also [19, Chapter 5]. Note that $\alpha_{1},\alpha_{2},\cdots,\alpha_{k+\varepsilon}=s$ are in fact the roots of the equation $M=L_{\tau}(n,s)$.

Remark 4.23**.**

The conditions for attaining the bounds (9) and (12) coincide. Thus a spherical $(n,M,s)$ code attains (9) if and only it attains (12). In particular, every tight spherical design attains (12).

Theorem 4.24** **([68] for (9), [26]

for (12)).

The bounds (9) and (12) cannot be improved by using in Theorem 4.7 and 4.19, respectively, polynomials of the same or lower degree.

Theorem 4.25** **([24] for (9),

[26] for (12)).

Let

[TABLE]

The bounds (9) and (12) (where $M=L_{\tau}(n,\alpha_{k+\varepsilon})$ for (12)) can be (simultaneously) improved if and only if $Q_{j}^{(n)}<0$ for some $j>\tau$.

Definition 4.26**.**

A spherical code $\mathcal{C}\subset\mathbb{S}^{n-1}$ is universally optimal if it (weakly) minimizes $h$-energy among all configurations of $|\mathcal{C}|$ points on $S^{n-1}$ for each absolutely monotone function $h$.

Theorem 4.27** ([32]).**

Every spherical code which is a spherical $(2k-1)$-design and which admits exactly $k$ inner products between distinct points is universally optimal. The 600-cell (the unique $(4,120,(1+\sqrt{5})/4)$-spherical code) is universally optimal.

See also [8, 19, 33].

Remark 4.28**.**

Any code which attains (9) (and (12)) is universally optimal. Is it unknown if there exists a spherical code, apart from the 600-cell, which is universally optimal but does not attain (9) (and (12)). Tables with all known codes which attain (9) (and (12)) can be found in [56, 58, 32].

Example 4.29**.**

Consider the case $(n,M)=(4,24)$. The well known code $D_{4}$ ($D_{4}$ root system; equivalently, the set of vertices of the regular $24$-cell) is optimal in the sense that it realizes the fourth kissing number [62]. However, this code is not universally optimal [30], despite having energy which is very close to the bound (12). For example, with the Newtonian $h(t)=\frac{1}{2(1-t)}$ it has energy $334$, while (12) gives $333$ (which can be improved to $\approx 333.157$).

Conjecture 4.30**.**

Every universally optimal spherical code attains the Linear Programming Bound of Theorem 4.19. **

Example 4.31**.**

(Example 3.25 continued.) A standard construction (see [35, Chapter 5]) maps binary codes from $\mathbb{F}_{2}^{n}$ to the sphere $\mathbb{S}^{n-1}$ – the coordinates [math] and $1$ are replaced by $\pm 1/\sqrt{n}$, respectively. Denote this map by ${\bf x}\to\overline{{\bf x}}$. The inner product $(\overline{{\bf x}},\overline{{\bf y}})$ on $\mathbb{S}^{n-1}$ and the Hamming distance ${\rm d_{H}}({\bf x},{\bf y})$ in $\mathbb{F}_{2}^{n}$ are connected by $(\overline{{\bf x}},\overline{{\bf y}})=1-\frac{2{\rm d_{H}}({\bf x},{\bf y})}{n}$. Thus the weights of the Kerdock code $\mathcal{K}_{\ell}$ correspond to the inner products $1,\frac{1}{\sqrt{n}},0,-\frac{1}{\sqrt{n}},-1$, respectively.

The image $\overline{\mathcal{K}}_{\ell}\subset\mathbb{S}^{2^{2\ell}-1}$ of $\mathcal{K}_{\ell}$ is asymptotically optimal with respect to both bounds (9) and (12). For example, it has energy

[TABLE]

When $n$ tends to infinity, we obtain

[TABLE]

which coincides with the asymptotic of (12) (obtained by a polynomial of degree 5).

5 Linear programming in other coding theory problems

Remark 5.1** (Linear programming in Johnson spaces).**

All concepts and results from Sections 1-3 hold true for the Johnson spaces with changes corresponding to the role of the Hahn polynomials instead of the Krawtchouk polynomials. See [36, 39, 56, 58, 59, 61].

Theorem 5.2** (Linear Programming Bound for binomial moments [5]).**

Let $\mathcal{C}\subset F_{2}^{n}$ be a code with distance distribution $(B_{0},B_{1},\ldots,B_{n})$, $w\in\{1,2,\ldots,n\}$, and

[TABLE]

Let the real polynomial $f(z)=\sum_{i=0}^{n}f_{i}K_{i}^{(n,2)}(z)$ satisfy the conditions

• (i)

$f_{i}\geq 0$* for $i=1,2,\ldots,n$;*

• (ii)

$f(j)\leq\binom{n-j}{n-w}$* for $j=1,2,\ldots,n$.*

Then $\mathcal{B}_{w}\geq f_{0}|\mathcal{C}|-f(0)$.

The final bound in this chapter is an upper bound on the size of a quantum code.

Theorem 5.3** (Linear Programming Bound for quantum codes [6]).**

*(see also [53])

Let $f(z)=\sum_{i=0}^{n}f_{i}K_{i}^{(n,4)}(z)$ be a polynomial satisfying the conditions*

• (i)

$f_{i}\geq 0$* for every $i=0,1,\ldots,n$;*

• (ii)

$f(z)>0$* for every $z=0,1,\ldots,d-1$;*

• (iii)

$f(z)\leq 0$* for every $z=d,d+1,\ldots,n$.*

Then every $((n,K))$ quantum code of minimum distance $d$ satisfies

[TABLE]

See also [7, 14, 15, 31, 41].

Acknowledgements. A significant part of the work of the first author on this chapter was done during his stay (August-December 2018) as a visiting professor at Department of Mathematical Sciences at Purdue University Fort Wayne. His research was supported, in part, by a Bulgarian NSF contract DN02/2-2016.