I Introduction
The problem of coded caching introduced in [1], plays a crucial role in reducing peak hour traffic in networks. A part of the content is made available in local cache of users so that traffic can be reduced at peak hours. Coded caching scheme involves two phases: a placement phase and a delivery phase. In the placement phase or the prefetching phase, which is performed during offpeak times, the entire database is made available to each user. Users fill their cache with the available data. Delivery phase is performed during peak traffic time. During placement phase some parts of files have to be judiciously cached at each user in such a way that the rate of transmission is reduced during the delivery phase. The prefetching can be done with or without coding. If during prefetching, no coding of parts of files is done, the prefetching scheme is referred to as uncoded prefetching [1, 2]. If coding is done during prefetching stage, then the prefetching scheme is referred to as coded prefetching [3, 4].
The seminal work in [1] shows that apart from the local caching gains obtained by placing contents at user caches before the demands are revealed, a global caching gain can be obtained by coded transmissions. This scheme is extended to decentralized scheme in [5]. More extensions to nonuniform demands [6] and online coded caching [7] are also available in literature.
If the shared bottleneck link between the server and the users is errorprone during the delivery phase, an error correcting delivery scheme is required. The minimum average rate and minimum peak rate of error correcting delivery schemes is characterized in [8]. The placement phase is assumed to be errorfree. This assumption can be justified as during placement phase there is no bandwidth constraint and any number of retransmissions can be done to make the placement errorfree. A similar model in which the delivery phase takes place over a packet erasure broadcast channel was considered in [9].
In this paper, we consider the coded caching problem considered in [3], where coded prefetching involves coding of parts of files. Optimal error correcting delivery scheme is proposed for this scheme. The main contributions of this paper are as follows.

An error correcting delivery scheme for coded caching problem with coded prefetching for small buffer sizes is proposed. We find expressions for average rate and peak rate of this error correcting delivery scheme (Section IV).
In this paper denotes the finite field with elements, where is a power of a prime, and denotes the set of all nonzero elements of . The notation is used for the set for any integer . For a matrix , denotes its
th row. For vector spaces
, denotes that is a subspace of .A linear code over is a dimensional subspace of with minimum Hamming distance . The vectors in are called codewords. A matrix of size whose rows are linearly independent codewords of is called a generator matrix of . A linear code can thus be represented using its generator matrix as, Let denote the length of the shortest linear code over which has dimension and minimum distance .
Ii Preliminaries and Background
To obtain error correcting delivery schemes we use results from error correction for index coding with coded side information. In this section we recall results from error correction for index coding with coded side information introduced in [10]. We also review the coded caching scheme with coded prefetching proposed in [3].
Iia Generalized Index Coding Problem and Error Correction
The index coding (IC) problem with sideinformation was introduced by Birk and Kol [11, 12]. A sender broadcasts messages through a noiseless shared channel to multiple receivers, each demanding certain messages and knowing some other messages a priori as sideinformation. The sender needs to meet the demands of each receiver in minimum number of transmissions. In [13] and [14], a generalization of the index coding problem was discussed, where the demands of the receivers and the sideinformation are linear combinations of the messages. In [14], the authors refer to this class of problems as Generalized Index Coding (GIC) problems.
An instance of GIC problem is described formally as follows. There is a message vector and there are receivers. The th receiver demands a linear combination of the messages , for some , where is the request vector and is the request packet of the th receiver. The sideinformation is represented by a matrix , where is the number of packets possessed as sideinformation by the th receiver. Though is unknown to the receiver , it can generate any vector in the row space of , denoted by . Let be an matrix over having as its th row. The matrix represents the demands of all the receivers. In the definition of GIC problem in [14] the source is assumed to possess only certain linear combinations of messages. In our work, it is assumed that all the messages are independent and the source possesses all of them.
The minrank of an instance of the GIC problem over is defined as
It is shown in [10] that the minrank is optimal length of linear generalized index code. For each , the set is defined as
(1) 
The set is defined as The maximum dimension of any element of is called the generalized independence number, denoted by . Thus dimension of any subspace of in serves as a lower bound for . It was shown in [10] that the minrank serves as an upper bound for the generalized independence number,
(2) 
Generalized index coding problems were classified in
[18]. In Generalized Index Coding with Coded SideInformation GIC (CSI) problems, demand of every receiver is uncoded but the sideinformation is coded. In Generalized Index Coding with Coded Demands GIC (CD) problems, the sideinformation of every receiver is uncoded but the demand is coded. In our work the focus is on GIC (CSI) problems.Error correcting index codes were introduced in [15] and later extended for generalized index coding problems in [10]. An Error Correcting Generalized Index Code (ECGIC) is a map that encodes the message vector such that each user, given its sideinformation and received transmissions with at most transmission errors, can decode its requested packet . An optimal linear ECGIC over is a linear ECGIC over of the smallest possible length . The length of an optimal linear ECGIC, satisfies
(3) 
where is the length of an optimal linear classical errorcorrecting code of dimension and minimum distance over [15, 10].
IiB Error Correcting Coded Caching Scheme
Error Correcting coded caching scheme was proposed in [8]. The server is connected to users through a shared link which is error prone. The server has access to files , each of size bits. Every user has an isolated cache with memory bits, where . A prefetching scheme is denoted by . During the delivery phase, only the server has access to the database. Every user demands one of the files. The demand vector is denoted by , where is the index of the file demanded by user . The number of distinct files requested in is denoted by . The set of all possible demands is denoted by During the delivery phase, the server informed of the demand , transmits a function of , over a shared link. Using the cache contents and the transmitted data, each user needs to reconstruct the requested file even if transmissions are in error.
For the error correcting coded caching problem, a communication rate is achievable for demand if and only if there exists a transmission of bits such that every user is able to recover its desired file even after at most transmissions are in error. Rate is the minimum achievable rate for a given , and . The average rate is defined as the expected minimum average rate given and under uniformly random demand. Thus
The average rate depends on the prefetching scheme . The minimum average rate is the minimum rate of the delivery scheme over all possible . The ratememory tradeoff for average rate is finding the minimum average rate for different memory constraints . Another quantity of interest is the peak rate, denoted by , which is defined as The minimum peak rate is defined as
IiC Coded Caching Scheme with Coded Prefetching
A coded caching scheme for small cache sizes involving coded prefetching was proposed in [3]. We call this scheme as Chen Fan Letaief (CFL) scheme. The system consists of a server and users. The server has access to files , each of size bits. Every user has an isolated cache with memory bits. The prefetching scheme is denoted by .

Consider the case when and . Each file is split into subfiles, i.e., During prefetching, the cache of user is designed to be , an XORed version of subfiles. It is shown in [3] that for and for are achievable. Furthermore if is achievable by memory sharing.

Consider and Each file is split into subfiles, i.e., The cache of user is given by for For the number of distinct demands files, it is shown in [3] that is achievable. For , the rate is achievable. Furthermore if is achievable by memory sharing.
For a fixed prefetching and for a fixed demand , the delivery phase of a coded caching problem is an index coding problem [1]. In fact, for fixed prefetching, a coded caching scheme consists of parallel index coding problems one for each of the possible user demands. Thus finding the minimum achievable rate for a given demand is equivalent to finding the minrank of the equivalent index coding problem induced by the demand .
Consider the CFL prefetching scheme . The index coding problem induced by the demand for CFL prefetching is denoted by Each subfile corresponds to a message in the index coding problem. Since prefetching is coded, represents a GIC (CSI) problem.
Iii Generalized Independence Number for
In this section we find a closed form expression for generalized independence number of the index coding problem . There are two different prefetching schemes employed in [3] depending upon the relationship between number of messages and number of receivers. For both these prefetching schemes, the generalized independence number of the corresponding index coding problem is shown to be equal to the minrank.
Iiia Number of files equal to number of users ()
In the CFL prefetching scheme, each file is split into subfiles. Hence the number of messages in is . Each user is split into receivers each demanding one message. Hence there are a total of receivers. From the expressions of the achievable rates in [3], we get the minrank as
(4) 
We find the generalized independence number for . The technique of obtaining is illustrated in the following example.
Example III.1
Consider a coded caching problem with , . Since , the CFL scheme is used for solving the coded caching problem. Each file is split into subfiles as , and . Let denote the vector obtained by concatenating and . The cache contents of user is for
For a given demand , this problem becomes the generalized index coding problem . We calculate the generalized independence number for this problem. For different demands, generalized independence number is calculated and it is shown to be equal to minrank of the corresponding generalized index coding problem.
First consider that all the demands are distinct, i.e., . Without loss of generality we can assume that the demand is Consider the equations Let be the subspace of , which consists of the vectors satisfying the equations and . From rank nullity theorem, dim. The induced generalized index coding problem has 9 messages and 9 receivers. For this case, (1) can be rewritten as Let . The generalized independence number is the maximum dimension of any subspace of in . We claim that all the vectors of belong to the set . This would mean . From the definition of , it is clear that the all zero vector belonging to also belongs to . Any other vector in will have at least one nonzero coordinate . The vector belonging to , having belongs to . Thus all vectors in lie in and From (4), we get . Hence by (2), we have
Consider the case when . Let Here we consider the same set of equations and and their solution space . Following the same argument as before, any vector in with for and lies in the corresponding set . From and , the condition forces at least one for . Thus vectors in with also lie in . Hence all vectors in lie in . Thus even in this case From (4), we get . Hence
Finally assume . Let In addition to and consider the following set of equations Let be the subspace of , which consists of the vectors satisfying the set of equations We follow the similar argument as above to show that all the vectors in lie in . By definition, lies in . All vector in with for are present in . By and , all the vectors in have . The condition and the set of equations force . Hence all vectors in with are present in . Thus all vectors in are present in . Moreover, dim Therefore From (4), . Thus by (2),
The example illustrates that the generalized independence number of the index coding problem is equal to its minrank. For different demands, the generalized index coding problem changes and for all those problems, minrank and generalized independence number are shown to be equal. This can be shown for all values of as given in the theorem below.
Theorem III.2
For and ,
where is the number of distinct demands.
Proof:
In CFL prefetching scheme , each file is split into subfiles . User caches . Let be the vector obtained by concatenation of vectors .
For a given demand , the delivery phase of the coded caching problem becomes a generalized index coding problem with messages and receivers.
First consider that all the demands are distinct, i.e., . Let the demand of the th user be . Thus . Consider the set of equations denoted by , where
Let be the subspace of which consists of the vectors satisfying the set of equations . From rank nullity theorem, we have dim.
For , from (1) we have, Let . The generalized independence number is the maximum dimension of any subspace of in . We show that is such a subspace. For this we need to show that all vectors of lie in . By definition of , the all zero vector lies in . Any other vector in will have at least one nonzero coordinate. The vectors belonging to having belongs to the set . Thus all vectors in lie in . The generalized independence number From (4), we get . Hence by (2), we have
Consider the case where . Without loss of generality we can assume that the first users have distinct demands and that the th user demands the file for . Without loss of generality, we can assume that the set of indices of the files that are not demanded are . There are files which are not demanded. In addition to , consider the following set of equations , for . The number of equations is thus . Let be the subspace of which consists of vectors satisfying these equations. Hence, dim By definition, lies in . Any vector with the coordinate for lies in . The set of equations force all for . Moreover if the set of equations force some for some . Hence any vector with lies in some for . Thus all vectors in lie in . Therefore Applying (4) and (2),
IiiB Number of users more than the number of files ()
In the CFL prefetching scheme for , each file is split into subfiles. Hence the number of messages in is . Each user is split into receivers in each demanding a single message. Thus there are a total of receivers. From the expressions for achievable rates in [3], we get the minrank as
(5) 
We find the generalized independence number for . The technique of obtaining is illustrated in the following example.
Example III.3
Consider a coded caching problem with , and . According to the CFL scheme each file is split into subfiles as , and . Let denote the vector obtained by concatenating and . The cache of the th user contains three coded packets for For a given demand , this problem becomes a generalized index coding problem having 36 messages and 48 receivers.
First consider that i.e., and Consider the equations given by for and . Thus there are nine equations. Let be the subspace of vectors in satisfying these nine equations. From rank nullity theorem, we get dim. For this case, (1) can be rewritten as for . Let . The generalized independence number is the maximum dimension of any subspace of in . We claim that is such a subspace. This would mean that . For this we need to show that all vectors in lie in . By definition of , the all zero vector lies in . Any other vector in will have at least one nonzero coordinate. All vectors in , having belongs to . Thus all vectors in lie in and From (5), we get . Hence by (2), we have
Consider now that and . In addition to the nine equations for and , consider three more equations for . Thus we consider a set of twelve equations given by . Let be the subspace of consisting of vectors which satisfy the equations in . Hence from rank nullity theorem, we have dim By definition, lies in . Any nonzero vector in with for lies in the corresponding . By , any forces some for and hence such vectors also lie in . Thus all vectors in lie in . Therefore From (5), we get . Hence by (2), we have
Finally consider and . The files and are not demanded by any user. In addition to the equations in , here we consider a set of equations for . Thus there are 24 equations in total. Let be the subspace of which satisfy these equations. By rank nullity theorem, the dimension of is given by dim The next step is to show that all the vectors in lie in . The all zero vector lies in by definition. Any nonzero vector in with for lies in the corresponding . From the set of equations, we have for . By , any forces for and hence such vectors also lie in . Thus all vectors in lie in . Therefore From (5), we get . Hence by (2), we have
The theorem below gives the expression for , when .
Theorem III.4
For and ,
where is the number of distinct demands.
Proof:
For and , the CFL prefetching scheme is as follows. Each file is split into subfiles . User caches coded packets given by for Let be the vector obtained by the concatenation of vectors For a given demand , this problem becomes a generalized index coding problem with messages and receivers.
First consider that all the demands are distinct, i.e., . Without loss of generality we can assume that the first users demand distinct files such that the th user demands for . Thus such that for . Let represent a set of equations, where . We consider a subset of the equations in of the form for . There are such equations. Let be the subspace of consisting of vectors satisfying these equations. From rank nullity theorem we have dim.
For , (1) can be rewritten as for and . Let . The generalized independence number is the maximum dimension of any subspace of in . We show that is such a subspace. For this we need to show that all vectors of lie in . By definition of , the all zero vector lies in . The vectors belonging to having belongs to the set . Thus all vectors in lie in and From (5), we get . Hence by (2), we have
Consider the case where . Let the first demands be distinct and the th user demands for Without loss of generality we can assume that the indices of the files which are not demanded are . There are files which are not demanded. In addition to the equations in , consider the following equations , for and . The number of equations is thus . Let be the subspace of which consists of the vectors satisfying these equations. By rank nullity theorem, dim By definition, lies in . Any vector in with the coordinate for lies in . The set of equations force all for and . Moreover by the set of equations in , would mean some other for . Hence any vector with lies in some for . Thus all vectors in lie in . Therefore From (4), we have . Hence from (2),
Iv Optimal Error Correcting Delivery Scheme for CFL Prefetching Scheme
In this section we give an expression for the average rate and worst case rate for a error correcting delivery scheme for CFL prefetching scheme. Also we propose a error correcting delivery scheme for this case. From Theorem III.2 and Theorem III.4, we can conclude that for all the generalized index coding problems induced from the CFL prefetching scheme,
(6) 
Hence, the and bounds in (3) meet. Using this the optimal error correcting delivery scheme can be constructed for CFL prefetching scheme and hence the average rate can be calculated as given in the following theorem.
Theorem IV.1
For a coded caching problem with CFL prefetching scheme for ,
where is the number of subfiles into which each file is divided in the CFL scheme. Furthermore, for , equals the lower convex envelope of its values at and .
Proof:
From (6) and (3), we can conclude that for any generalized index coding problem induced from the coded caching problem with CFL prefetching, the and bounds meet. Thus the optimal error correcting delivery scheme would be the concatenation of the CFL delivery scheme with an optimal linear error correcting code. The optimal length or equivalently the optimal number of transmissions required for error corrections in those generalized index coding problems is thus and hence the statement of the theorem follows for . For
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