### Contributions
* Technical novelty: To obtain the proposed inner bound on the DKI capacity region, we address the input set imposed by the input constraints, and exploit it for an appropriate ball covering (overlapping balls with identical radius); namely, we consider covering of hyper balls inside a Hamming cube, whose Hamming radius grows in the codeword length \(n\), i.e., \(\sim\!\!n\beta\), for some \(\beta\in(0,1)\) upper bounded by a function of the channel statistic. We exploit a greedy construction similar as for the _Gilbert bound_ method. While the radius of the small balls in the DI problem for the Gaussian channel with slow and fast fading [32], tends to zero as \(n\to\infty\), here, the radius similar to the DKI problem for the slow fading channel [51] grows in the codeword length \(n\) for asymptotic \(n\). In general, the derivation of lower bound for the BSC is more complicated compared to that for the Gaussian [32] and Poisson channels with/out memory [48; 52], and entails exploiting of new analysis and inequalities. Here, the error analysis in the achievability proof requires dealing with several combinatorial arguments and using of bounds on the tail for the cumulative distribution function (CDF) of the Binomial distribution. The DKI problem was recently investigated in [52] for a DTPC with ISI where the size of the ISI taps is assumed to scale as \(L(n,l)=2^{l\log n}\). In contrast to the findings in [52], where the attainable rate region of triple rates \((\kappa,l,R)\) for the Poisson channel with memory was derived, here, we study the DKI problem for a memoryless BSC, i.e., \(L=1\), and the attainable rate region of pair rates \((\kappa,R)\) is established. Furthermore, while the method in the achievability proof of [52] is based on sphere packing, which includes an arrangement of _non-overlapping_ spheres in the feasible input set. Here, we use a rather different approach called sphere/ball covering, which allows for the spheres/ball to _overlap_ with each other. For the derivation of the outer bound on the DKI capacity region, it is assumed that a random series of code with diminishing error probabilities is provided. Then, for such a sequence, we prove that an one-to-one mapping between the message set and the set of the feasible input set (induced by the input constraint) can be established. Unlike the previous upper bound proof for DI over the DMC [32]; here, the proof for corresponding lemma is adopted in order to incorporate _relevant_ set of the goal message sets, appropriately. Moreover, in the converse proof, similarly to [52], the method of proof by contradiction was utilized; that is, assuming that a certain property regarding the distance or number of the codewords is negated, we lead to a contradiction related to the sum of the sort I and sort II error probabilities. However, unlike [52], where a sub-linear function for the size of the goal message set was considered, i.e., \(K(n,\kappa)=2^{\kappa\log n}=n^{\kappa}\), here, our converse entails a faster function, namely \(K(n,\kappa)=2^{\kappa n}\). **Notations:** We use the subsequent notations throughout this paper: We use symbol \(\triangleq\) for a definition. Alphabet sets are shown by blackboard bold letters \(\mathsf{K},\mathsf{X},\mathsf{Y},\mathbb{Z}\).... Random variables (RVs) are indicated by upper case letters \(X,Y,Z,\ldots\). Constants and values (realization) of RVs are specified by lower case letters \(x,y,z,\ldots\). Row vectors of size \(n\), i.e., \(\mathbf{x}=(x_{1},\ldots,x_{n})\) and \(\mathbf{y}=(y_{1},\ldots,y_{n})\), are represented by lower case bold symbol \(\mathbf{x}\) and \(\mathbf{y}\). The distribution of a RV \(X\) is specified by a probability mass function (pmf) \(p_{X}(x)\) over a finite set \(\mathcal{X}\). The CDF of a Binomial RV is indicated by \(B_{X}(x)\triangleq\Pr(X\leq x)\). All information quantities and logarithms are in base 2. Symbol \(\llbracket M\rrbracket\) represents the set of all consecutive natural numbers from 1 to \(M\). We indicate the modulo two addition operator by \(\oplus\).