A. Motivation is given by (3). We now state a multishot counterpart for the result of Silva et.al presented in Section II. Theorem 1: Let X be a multishot matrix code. If d$_{S}$ ( 〈 X 〉 ) > 2(2 ρ + τ ) then X is ( ρ, τ ) -correcting. Proof: The proof is a simple generalization of [2, The- orem 1]. Let X = ( X $_{0}$, . . . , X n − $_{1}$) be the transmitted code- word and Y = ( Y $_{0}$, . . . , Y n − $_{1}$) be the received sequence, according to (5). Let 〈 X 〉 = ( 〈 X $_{0}$〉 , . . . , 〈 X n − $_{1}$〉 ) and 〈 Y 〉 = ( 〈 Y $_{0}$〉 , . . . , 〈 Y n − $_{1}$〉 ) . We have One of the basic problems in the realm of one-shot matrix coding is to find codes with good rates and good error- correcting capabilities. To achieve both goals simultaneously, it may be unavoidable to increase the field size, q , or the packet size, T . Multishot codes allow for a third possibility: to increase the number of channel uses, n . With that in mind, multishot codes are attractive when the system under consideration is such that it is not possible to change the field and packet size. This is true, for example, in fast-topology changing networks (such as wireless ones), where the transfer matrix doesn’t stay the same for much long. Under this circumstance, to obtain codes with better error-correcting capabilities we must spread redundancy across multiple shots. Put another way, when errors occur in a random fashion and q and T are fixed, we must use the matrix channel many times in order to approach the channel capacity. For a simple example in which a multishot code is capable of detecting more errors when compared with the best one could do by simply repeating one-shot codes, we point the reader to [5]. Since ρ + 2 τ < d$_{S}$ ( 〈 X 〉 ) / 2 , a minimum extended subspace distance decoder is guaranteed to yield 〈 X 〉 given 〈 Y 〉 . In this work we adopt a block-coding approach, in which the matrix channel (1) is used n times in a row. Our channel model then becomes Let u = ( u $_{0}$, . . . , u n − $_{1}$) and v = ( v $_{0}$, . . . , v n − $_{1}$) be two n - tuples of vectors in F N q M . The extended rank distance between them is defined by d$_{S}$ ( 〈 X 〉 , 〈 Y 〉 ) = n − 1 ∑ j =0 d$_{S}$ ( 〈 X $_{j}$〉 , 〈 Y $_{j}$〉 ) ≤ n − 1 ∑ j =0 d$_{S}$ ( 〈 X $_{j}$〉 , 〈 A j X $_{j}$〉 ) +$^{n}$$^{−}$$^{1}$ ∑ j =0 d$_{S}$ ( 〈 A j X $_{j}$〉 , 〈 Y $_{j}$〉 ) ≤ n − 1 ∑ j =0 rankdef A j + 2$^{n}$$^{−}$$^{1}$ ∑ j =0 rank Z j ≤ ρ + 2 τ. Y j = A j X j + Z $_{j}$, (5) d$_{R}$ ( u , v ) = n − 1 ∑ j =0 d$_{R}$ ( u $_{j}$, v $_{j}$) , n − 1 ∑ j =0 rankdef A j ≤ ρ and n − 1 ∑ j =0 rank Z j ≤ τ. (6) d$_{S}$ ( 〈I ( u ) 〉 , 〈I ( v ) 〉 ) = 2 d$_{R}$ ( u , v ) . R ( X ) = log$_{q}$ | X | nN T ; D. Multishot Rank-Metric Codes B. Model and Definitions where d$_{R}$ ( · , · ) in the right-hand side is the rank distance as defined in (4). Just like regular rank-metric codes, multish ot rank-metric codes can be applied to noncoherent network coding. To this end, we use an extended version of the lifting operation defined as I ( u ) = ( I ( u $_{0}$) , . . . , I ( u n − $_{1}$)) , where u = ( u $_{0}$, . . . , u n − $_{1}$) . It is straightforward to show that for j = 0 , . . . , n − 1 , with matrices $^{X}$$_{j}$, $^{Y}$$_{j}$, $^{A}$$_{j}$, and Z j retaining their dimensions from the one-shot case. We allow the adversities to be spread in any way among the n time slots: Thus, a multishot rank-metric code R ⊆ ( F N q M ) n gives rise to a multishot matrix code X ⊆ ( F N × T q ) n (where T = N + M ) defined by X = I ( R ) = {I ( u ) : u ∈ R } , with | X | = | R | and d$_{S}$ ( 〈 X 〉 ) = 2 d$_{R}$ ( R ) . An n - shot (or multishot ) matrix code X is a non-empty subset of ( F N × T q ) $^{n}$. Its code rate is defined as the ratio between the amount of information symbols conveyed by the transmission of a codeword and the amount of physical symbols spent by each codeword, that is, The multilevel code construction was proposed by Imai and Hirakawa [6] in 1977 and became very popular in the 80’s and 90’s with more general constructions being developed by many other researchers. Although originally targeted at codes over a given signal set S of the Euclidean space, the construction can be generalized for block codes over any finite subset S of a given metric space M with associated distance d$_{M}$ ( · , · ) . (This is true as long as the component codes are Hamming-metric.) Next, we base our description of the multilevel construction on the work of Lin and Costello [8, Chapter 19], wherein many references on this subject are listed. Given a set S , an m -level partitioning of S is defined by a sequence of m + 1 partitions Γ$_{0}$ , . . . , Γ$_{m}$ of S such that where U = ( U$_{0}$, . . . , U$_{n}$$_{−}$$_{1}$ ) and V = ( V$_{0}$, . . . , V$_{n}$$_{−}$$_{1}$ ) are n - tuples of subspaces of F N q and d$_{S}$ ( · , · ) in the right-hand side we have 0 ≤ R ( X ) ≤ 1 . Akin to the one-shot case, a multishot code X is said to be ( ρ, τ ) -correcting if ( X $_{0}$, . . . , X n − $_{1}$) can be unambiguously determined from ( Y $_{0}$, . . . , Y n − $_{1}$) for all choices of ( A $_{0}$, . . . , A n − $_{1}$) and ( Z $_{0}$, . . . , Z n − $_{1}$) subject to (6). We extend the subspace distance to IV. GENERAL MULTILEVEL CODE CONSTRUCTION C. The Extended Subspace Distance d$_{S}$ ( U , V ) = n − 1 ∑ j =0 d$_{S}$ ( U$_{j}$, V$_{j}$ ) ,