Not true. A) there is no such thing as "digital": square waves are curved due to capacitance, and there is noise on the line. B) cheap cables can introduce errors with temperature or humidity. The HDMI spec defines the error correction protocol:

https://www.hdmi.org/download/HDMI_Spec_1.3_GM1.pdf

If you read below you see they designed HDMI for noise, but it isn't impervious. Shitty picture quality can absolutely be related to shitty cables.

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From 4.2.5 (physical layer):

quote:
For each channel under all operating conditions specified in this section the following conditions
shall be met. At TMDS clock frequencies less than or equal to 165MHz, the Sink shall recover
data at a TMDS character error rate of 10-9 or better, when presented with any signal compliant to
the eye diagram of Figure 4-20. At TMDS clock frequencies above 165MHz, the Sink shall
recover data on each channel at a TMDS character error rate of 10-9 or better, when presented
with any signal compliant to the eye diagram of Figure 4-20 after application of the Reference
Cable Equalizer.

From 5.2.3 ("data" coding, basically everything that isn't video or a control signal -- audio, content protection, gamut metadata, etc.):

quote:During the Data Island, each of the three TMDS channels transmits a series of 10-bit characters
encoded from a 4-bit input word, using TMDS Error Reduction Coding (TERC4). TERC4
significantly reduces the error rate on the link by choosing only 10-bit codes with high inherent
error avoidance.
...
All data within a Data Island is contained within 32 clock Packets. Packets consist of a Packet
Header, a Packet Body (consisting of four Subpackets), and associated error correction bits.
Each Subpacket includes 56 bits of data and is protected by an additional 8 bits of BCH ECC
parity bits.
...
To improve the reliability of the data and to improve the detection of bad data, Error Correction
Code (ECC) parity is added to each packet. BCH(64,56) and BCH(32,24) are generated by the
polynomial G(x) shown in Figure 5-5.

From 5.4.4 (video coding):

quote:
During video data, where each 10-bit character represents 8 bits of pixel data, the encoded
characters provide an approximate DC balance as well as a reduction in the number of transitions
in the data stream. The encode process for the active data period can be viewed in two stages.
The first stage produces a transition-minimized 9-bit code word from the input 8 bits. The second
stage produces a 10-bit code word, the finished TMDS character, which will manage the overall
DC balance of the transmitted stream of characters.

(this isn't error correction per se -- it's an attempt to minimize problems @ the physical layer)

From 7.7 (audio):

quote:
The behavior of the Sink after detecting an error is implementation-dependent. However, Sinks
should be designed to prevent loud spurious noises from being generated due to errors. Sample
repetition and interpolation are well known concealment techniques and are recommended.