Никому не сообщать код.

Because сообщать takes the person you tell in the dative case.

• никто = no one as a subject
• никому = to no one / to nobody

In this sentence, no one is not doing the action. Instead, the code must not be communicated to anyone, so Russian uses the dative:

• сообщать кому? = to tell whom?
• никому = to nobody

So:

• Никто не сообщает код = No one is telling the code
• Никому не сообщать код = Do not tell the code to anyone

Russian often uses the infinitive for firm instructions, warnings, rules, and official notices.

So Никому не сообщать код sounds like:

• a written instruction
• a policy
• a warning
• something you might see on a screen, in a manual, or in a security message

Compare:

• Не сообщайте никому код. = Don’t tell anyone the code.
  More like a direct command to a person or group.

• Никому не сообщать код. = Do not disclose the code to anyone.
  More impersonal, formal, and instructional.

English does something similar in signs and instructions, like Do not enter or Not to be used.

Yes, никому already has a negative meaning, but in Russian this normally goes together with не on the verb. This is standard Russian grammar.

This is called negative concord: multiple negative elements appear together, but the sentence still has just one negative meaning.

So Russian says:

• никому не сообщать
• literally: to no one not to communicate

But the real meaning is simply:

• do not tell anyone

You should expect this pattern with other negative pronouns too:

• никого не вижу = I don’t see anyone
• ничего не знаю = I don’t know anything
• никогда не говори = never say

Because код is the direct object of сообщать, so it is in the accusative case.

For an inanimate masculine noun like код, the accusative singular is the same as the nominative singular:

• nominative: код
• accusative: код

So:

• сообщать код = to communicate the code / to tell the code

If the noun were feminine, you would often see a different accusative form:

• сообщать информацию = to communicate information

This is about aspect.

• сообщать = imperfective
• сообщить = perfective

In negative instructions, Russian often prefers the imperfective infinitive to mean do not do this at all / this action is prohibited.

So:

• Никому не сообщать код = Do not disclose the code to anyone
• it sounds like a standing rule or prohibition

If you used не сообщить, it would sound much less natural here. The perfective infinitive is not the normal choice for this kind of written warning.

As a broad rule:

• imperfective in prohibitions often focuses on banning the action in general
• perfective often points more to a single completed result

Yes, Код никому не сообщать is also possible.

Russian word order is more flexible than English, and changing the order changes the focus more than the basic meaning.

Compare:

• Никому не сообщать код.
  Emphasizes никому first: to no one

• Код никому не сообщать.
  Emphasizes код first: the code is what must not be disclosed

In a security context, Код никому не сообщать is actually very natural because it puts the sensitive item first.

Yes, that is grammatically correct too.

All of these are possible:

• Никому не сообщать код
• Не сообщать никому код
• Код никому не сообщать

The difference is mainly emphasis and style.

• starting with никому highlights to no one
• starting with не сообщать highlights the prohibition itself
• starting with код highlights the thing being protected

Russian often places the most important or contrastive information earlier in the sentence.

Сообщать код is correct and natural, especially in formal or security-related language. It means to communicate / disclose / tell the code.

Depending on context, Russians might also use:

• говорить код = to say the code
• называть код = to state the code
• передавать код = to pass on the code
• разглашать код = to divulge the code (more formal/serious)

But сообщать works very well for instructions like passwords, access codes, PINs, and verification codes.

Yes. The subject is not stated, because this kind of instruction is meant for whoever is reading it.

So the implied meaning is something like:

• You must not tell anyone the code
• One must not disclose the code to anyone

This omission is very common in Russian notices and instructions. The sentence is intentionally impersonal, which makes it sound more official and general.