For a convolutional code in the presence of a symbol erasure channel, the information debt $I(t)$ at time $t$ provides a measure of the number of additional code symbols required to recover all message symbols up to time $t$. Information-debt-optimal streaming ($i$DOS) codes are convolutional codes which allow for the recovery of all message symbols up to $t$ whenever $I(t)$ turns zero under the following conditions; (i) information debt can be non-zero for at most $\tau$ consecutive time slots and (ii) information debt never increases beyond a particular threshold. The existence of periodically-time-varying $i$DOS codes are known for all parameters. In this paper, we address the problem of constructing explicit, time-invariant $i$DOS codes. We present an explicit time-invariant construction of $i$DOS codes for the unit memory ($m=1$) case. It is also shown that a construction method for convolutional codes due to Almeida et al. leads to explicit time-invariant $i$DOS codes for all parameters. However, this general construction requires a larger field size than the first construction for the $m=1$ case.

翻译：对于存在符号擦除信道的卷积码，时间$t$处的信息债务$I(t)$衡量了恢复截至时间$t$的所有消息符号所需的额外码符号数量。信息债务最优流（$i$DOS）码是一类卷积码，其允许在以下条件下每当$I(t)$归零时恢复截至时间$t$的所有消息符号：（i）信息债务最多连续$\tau$个时隙不为零；（ii）信息债务从未超过特定阈值。已知所有参数下均存在周期时变$i$DOS码。本文研究了显式时不变$i$DOS码的构造问题。我们针对单位内存（$m=1$）情形给出了显式时不变$i$DOS码的构造方法。同时证明，Almeida等人提出的卷积码构造方法可用于生成所有参数下的显式时不变$i$DOS码。然而，相较于$m=1$情形的第一种构造，该通用构造需要更大的域规模。